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Downloaded from http://rsta.royalsocietypublishing.org/ on June 18, 2017 Phil. Trans. R. Soc. A (2012) 370, 3114–3128 doi:10.1098/rsta.2011.0556 REVIEW Numerical models of sunspot formation and fine structure B Y M ATTHIAS R EMPEL* High Altitude Observatory, National Center for Atmospheric Research, PO Box 3000, Boulder, CO 80307, USA Sunspots are central to our understanding of solar (and stellar) magnetism in many respects. On the large scale, they link the magnetic field observable in the photosphere to the dynamo processes operating in the solar interior. Properly interpreting the constraints that sunspots impose on the dynamo process requires a detailed understanding of the processes involved in their formation, dynamical evolution and decay. On the small scale, they give an insight into how convective energy transport interacts with the magnetic field over a wide range of field strengths and inclination angles, leading to sunspot fine structure observed in the form of umbral dots and penumbral filaments. Over the past decade, substantial progress has been made on both observational and theoretical sides. Advanced ground- and space-based observations have resolved, for the first time, the details of umbral dots and penumbral filaments and discovered similarities in their substructures. Numerical models have advanced to the degree that simulations of entire sunspots with sufficient resolution to resolve sunspot fine structure are feasible. A combination of improved helioseismic inversion techniques with seismic forward modelling provides new views on the subsurface structure of sunspots. In this review, we summarize recent progress, with particular focus on numerical modelling. Keywords: magnetohydrodynamic; solar magnetic field; sunspots 1. Introduction Sunspots are the most prominent manifestation of large-scale magnetic field on the visible solar surface. Understanding how sunspots form, evolve and decay is central to our understanding of solar magnetism, solar variability and its impact on the heliosphere. The solar magnetic field is produced in the convection zone (including the underlying overshoot region) through a magnetohydrodynamics (MHD) dynamo process [1]. Most of the observational constraints on the magnetic field that the dynamo produces come from photospheric observations, mostly in the form of active regions (sunspot groups) and their temporal behaviour throughout the solar magnetic cycle [2]. Unravelling the detailed implications for the magnetic field within the solar convection zone requires a detailed *[email protected] One contribution of 11 to a Theme Issue ‘Astrophysical processes on the Sun’. 3114 This journal is © 2012 The Royal Society Downloaded from http://rsta.royalsocietypublishing.org/ on June 18, 2017 Review. Numerical sunspot models 3115 understanding of the processes involved in the transport of the magnetic field towards the visible solar surface and formation of active regions and sunspots in the photosphere [3,4]. While the process of flux emergence is formally a part of the dynamo itself, it is typically modelled independently owing to the vast difference in the time and length scales involved. Substantial progress in modelling capabilities over the past decade has enabled a transition towards three-dimensional models incorporating the effects of convection down to scales of granulation in the solar photosphere. With these improvements, a self-consistent description of flux emergence across scales from the base of the convection zone into the solar photosphere and beyond is in reach within the next decade. While magnetic field and associated flows in the bulk of the solar convection zone are difficult to infer from surface observations, this task is much more promising for the uppermost 10–20 Mm of the convection zone where substantial progress was made through the application of local helioseismology [5–7]. Progress in this area thrives from a combination of stable, high-resolution space-based observations, advanced inversion methods and most recently also numerical forward modelling of wave propagation through sunspots. In particular, the latter promises to resolve many of the challenges that currently restrict the reliable use of helioseismic inversions within and near sunspots. The fine structure of sunspots present at the smallest currently observable scales has puzzled observers for more than a century (see recent reviews [8–12]). In the umbra of sunspots, fine structure is manifest as dot-like plumes with a diameter of several hundred kilometres (umbral dots), and in the penumbra as elongated filaments in which strong horizontal outflows (first discovered about a century ago [13]) along almost horizontal fields are embedded in a background of more vertically inclined field (see also figure 1 for fine structure as it is present in a numerical sunspot model). Sunspot fine structure is the consequence of magnetoconvective energy transport in strongly magnetized plasma, and therefore related to a fundamental astrophysical process. Over the past decade, the physical understanding of magneto-convection in sunspots was advanced forward by a combination of new instrumentation (both ground- and space-based) and realistic MHD models, which finally progressed to the point at which entire sunspots can be modelled with a resolution sufficient for capturing the essence of sunspot fine structure. In this review, we will present a selective summary of these developments with a special emphasis on recent advances in MHD modelling capabilities. After a brief review of recent developments in ground- and space-based observation in §2, we focus on MHD models of sunspot fine structure and of sunspot formation and flux emergence in §3. Progress in our understanding of the subsurface structure of sunspots from local helioseismology is presented in §4. Concluding remarks are given in §5. 2. Recent advances and future directions in observations Over the past decade, several advancements in ground- and space-based observations have pushed our understanding of sunspot fine structure forward. Substantial improvements in ground-based observations resulted from the development of advanced image correction (adaptive optics) and reconstruction Phil. Trans. R. Soc. A (2012) Downloaded from http://rsta.royalsocietypublishing.org/ on June 18, 2017 3116 M. Rempel I Bz Vz VR Figure 1. Sunspot fine structure at the t = 1 level. Shown are (in clockwise order) the intensity (I ), vertical magnetic field (Bz ), radial flow (with respect to centre of sunspot) (VR ) and vertical flow velocity (Vz ). The range shown is 0.3 . . . 1.5I for I (with I being the quiet sun intensity), ±2.5 kG (1 kG = 0.1 T) for Bz , ±8 km s−1 for VR and ±2 km s−1 for Vz . methods that pushed existing telescopes to their diffraction limit. Examples for ground-based observations of sunspot fine structure under optimal conditions have been described in previous studies [14–26]. While the filamentary structure of the penumbra was already known from lower-resolution observations, recent high-resolution data revealed an additional substructure of penumbral filaments in terms of dark cores embedded in bright filaments [14], which has also been confirmed for umbral dots [21,27] and light bridges and provides strong observational evidence for a common origin of these structures. Ground-based instrumentation is advanced even further with the recently deployed 1.6 m New Solar Telescope telescope at the Big Bear Solar Observatory as well as the 1.5 m Gregor telescope on Tenerife, which will become operational soon. New 4 m class telescopes such as the Advanced Technology Solar Telescope as well as the European Solar Telescope will become available over the next decade. A major step forward in space-based instrumentation is the Hinode satellite—in particular, the spectro-polarimeter of the Solar Optical Telescope [28]. Examples of recent results concerning sunspot fine structure are Phil. Trans. R. Soc. A (2012) Downloaded from http://rsta.royalsocietypublishing.org/ on June 18, 2017 Review. Numerical sunspot models 3117 those of Ichimoto et al. [29,30], Franz & Schlichenmaier [31] and Bharti et al. [32]. The recently launched Solar Dynamics Observatory (SDO) will further advance the study of sunspots and active regions through full-disc high-resolution magnetograms and Doppler data for helioseismic inversions. 3. Recent advances in magnetohydrodynamics modelling (a) General considerations Challenges for MHD models of the solar convection zone, including the photosphere, arise primarily from the vast separation in length and time scales that are the consequence of the strong stratification of the system, while the physical ingredients are well known. In the bulk of the convection zone, singlefluid MHD, combined with an ideal gas equation of state, is sufficient—when approaching the photosphere, the equation of state has to take into consideration partial ionization. In the photosphere, full three-dimensional radiative transfer is essential, but remains manageable in complexity owing to the validity of local thermal equilibrium, and the fact that the frequency dependence of opacities can be captured sufficiently well in most cases by either grey or multi-band approaches. Many of these approximations were introduced by Nordlund [33] and have been reviewed recently in the study of Nordlund et al. [34], see also the contribution by Stein [35]. In that sense, the solar convection zone and lower photosphere differ substantially from the solar chromosphere and corona, where, in addition to a large-scale separation, more complicated physics such as non-local thermodynamic equilibrium radiative transfer, multi-fluid MHD, non-equilibrium ionization, and kinetic effects matter. Nevertheless, the density variation by about six orders of magnitude within the convection zone implies a transition from strongly subsonic motions near the base of the convection zone (Mach numbers ∼10−4 ) to supersonic motions in the photosphere and a variation of the pressure scale height from about 50 Mm to 100 km. As a consequence, a model of the whole convection zone up into the photosphere is currently still out of reach. (b) Modelling of sunspot fine structure Most aspects of sunspot fine structure can be addressed by models that are restricted to the uppermost few megametres of the convection zone. Nevertheless, substantial challenges remain owing to the massive scale separation between the typical size of a sunspot (greater than 10 Mm) and the observed scale of fine structure smaller than a few hundred kilometres, which requires grid resolutions as small as a few 10 km. As a consequence, magneto-convection in strong magnetic fields was addressed initially mainly through idealized MHD simulations neglecting effects from partial ionization and radiative transfer. ‘Realistic’ MHD simulations considering those effects were initially applied to small sections of sunspots, and only in the past few years have computing resources been allowed for simulations of complete sunspots. In addition to computing resources, these simulations also require robust numerical schemes that can equally well handle low and high Mach number flows, as well as high- and low-plasma beta regimes. Phil. Trans. R. Soc. A (2012) Downloaded from http://rsta.royalsocietypublishing.org/ on June 18, 2017 3118 M. Rempel The primary goal of magneto-convective sunspot models is to provide a consistent description that explains the brightness of umbra and penumbra, the fine structure (umbral dots and penumbral filaments) and the origin of the Evershed flow. Apart from an initial setup and boundary conditions that define the overall field geometry of the sunspot, such models should reproduce all the other aspects that are self-consistently based on MHD and radiative transfer, provided that a sufficient numerical resolution is used. (i) Idealized models The first two- and three-dimensional MHD models addressed magnetoconvection in idealized setups wherein the researchers studied the influence of field strength and inclination on convection patterns. A central parameter in these studies is the ratio of magnetic to thermal diffusivity, z, which characterizes the mode of convection. Of particular interest is the situation with z < 1, which is realized in the upper-most parts of the convection zone. Here, it was found that the energy is transported by oscillatory columnar convection [36–39], which transitions to travelling waves in the presence of inclined magnetic fields [40,41]. (ii) Magnetohydrodynamics models of umbral dots A substantial step forward in realism was presented by Schüssler & Vögler [42], who modelled a sunspot umbra using MHD simulations with three-dimensional radiative transfer and a realistic equation of state. Here, it was found that umbral dots are quasi-stationary plumes (lifetimes of about half an hour) with overturning convection and substantially reduced field strength that are embedded in the vertical background field of the umbra. Interestingly, the oscillatory convection mode found in previous idealized investigations was not present. These simulations also predicted a substructure in terms of dark lanes within umbral dots similar to those that were identified previously in penumbral filaments by Scharmer et al. [14]. Observational confirmations for dark lanes in umbral dots were given by Bharti et al. [27] and Rimmele [21]. (iii) Magnetohydrodynamics models of penumbral filaments The first MHD simulations with radiative transfer addressing the structure of the penumbra were introduced by Heinemann et al. [43] and later refined by Rempel et al. [44]. To keep the computational expense moderate, these simulations were using a ‘slab-geometry’, in which the computational domain is restricted to a narrow cut through the centre of a sunspot and horizontal periodicity is assumed. Relaxing these constraints, Rempel et al. [45] presented a numerical MHD simulation of a pair of full sunspots, which provided, for the first time, extended penumbrae with strong horizontal outflows (Evershed flow). Results from a very recent high-resolution simulation [46,47] are presented in figure 1. The computational domain with an extent of 49 × 49 × 6.1 Mm3 is resolved with a total of 4.8 billion grid points (3072 × 3072 × 512). In clockwise order, figure 1 presents (bolometric) intensity, vertical magnetic field strength, radial and vertical flow velocity. Here, radial refers to the cylindrical radius with respect to the centre of the sunspot. The simulation shows umbral dots Phil. Trans. R. Soc. A (2012) Downloaded from http://rsta.royalsocietypublishing.org/ on June 18, 2017 Review. Numerical sunspot models 3119 in the centre and the filamentary structure of the penumbra with dominantly radially aligned structures present in all four quantities shown. Throughout most of the penumbra, bright filaments are associated with radially elongated convection cells having upflows in their centre and downflows in their periphery. Unlike granulation, where downflows have typically a stronger velocity amplitude, upflows dominate in the inner penumbra and possibly lead to an observational bias towards upflows. This might explain the lack of detection of a convective flow pattern in some observations [23,31]. A recent analysis of a numerical sunspot model by Bharti et al. [48] confirmed that most of the downflows remain hidden after the data is degraded to the resolution of typical observations. The horizontal divergence of upflowing material strongly reduces the vertical magnetic field, while the horizontal field (not shown in figure 1) is moderately enhanced, leading to a complex magnetic structure consisting of filaments with an almost horizontal field embedded in a more vertically inclined background field. Fast horizontal outflows with flow velocities close to the Alfvén velocity are found in the horizontal field component, consistent with the observationally inferred structure (see reviews [8–12]). The three-dimensional structure available from the numerical simulations allows us to further understand processes that are intrinsically hidden from photospheric observations. We highlight here three aspects. Mass and energy transport in penumbra. The brightness of sunspot penumbrae is typically more than 70 per cent of the brightness of granulation, indicating that a substantial amount of overturning convective motions should be present. Interestingly, the observational evidence for overturning motions in a sunspot penumbra is not that clear. While there is some evidence for overturning convection [16,20,21,30,32,49], other studies such as [23,29,31,50] see mostly support for a larger-scale flow pattern with upflows in the inner and downflows in the outer penumbra. Recent MHD models [51] clearly show vigorous convection in the penumbra and a relationship of the form I ∝ vz r.m.s. (t = 1), i.e. a brightness of 0.7I requires about 1 km s−1 vertical r.m.s. velocity, which is about half the value found in granulation. Direct evidence for overturning convection was found recently by Joshi et al. [24] and Scharmer et al. [25] using observations in the rather deep forming CI 5380 line. Penumbral fine structure. The complicated fine structure of the penumbra is a direct consequence of overturning convective motions. The vertical magnetic field component is strongly weakened in upflows owing to the horizontal divergence of fluid motions. At the same time, the horizontal field component shows a moderate enhancement within filaments near an optical depth of unity, which results from a combination of induction from the vertical shear profile of the Evershed flow with upward advection that concentrates horizontal flux near the optical depth of unity, where most of the mass flux turns over [51]. The result is a fine structure consisting of essentially horizontal fields within flow channels, embedded in a background of more vertically inclined fields. The combination of a reduction of vertical and moderate enhancement of horizontal field components leads to an overall reduction of field strength within flow channels in the inner penumbra and an increase in the outer penumbra. Origin of the Evershed flow. Since its discovery by Evershed [13], the origin of fast outflows in sunspot penumbrae has been central to almost all theoretical and observational investigations of sunspot fine structure. While many simplified Phil. Trans. R. Soc. A (2012) Downloaded from http://rsta.royalsocietypublishing.org/ on June 18, 2017 3120 M. Rempel models considered flows channelled by magnetic field in magnetic flux tubes, the large degree of overturning motions found in simulations limits the meaningful applicability of the flux tube concept in the case of penumbral flow channels. Based on the work of Heinemann et al. [43], it was proposed by Scharmer et al. [52] that the Evershed flow is essentially the horizontal convective flow component along the field, enhanced in amplitude due to anisotropy. A recent analysis by Rempel [51] showed that the Evershed flow originates from a narrow boundary layer forming in upflow regions just beneath the t = 1 level. Here, the horizontal magnetic field component increases strongly with height, leading to a sharp kink in field lines. Pressure-driven upflows provide the energy source for the horizontal acceleration of the Evershed flow, with the Lorentz force facilitating the energy exchange between the vertical and horizontal direction. The consequence is a magnetized horizontal outflow in the deep photosphere, with a flow amplitude close to the Alfvén velocity (5–10 km s−1 ). (iv) Future challenges MHD simulations with radiative transfer and realistic equations of state have evolved to the point where they include all essential building blocks of sunspot fine structure. All aspects of sunspot fine structure from umbral dots to penumbral filaments and light-bridges are explained through a common magnetoconvective process that is modulated by the background field strength and inclination angle. Large-scale properties, such as the flux and overall extent of penumbrae, is, in most simulations, a consequence of initial and boundary conditions; however, details of fine structure are the consequence of (mostly) resolved magneto-convection processes that are found to be robust within the currently accessible resolution range. While the overall flow and field structure at photospheric levels agrees well with the observationally inferred fine structure, a detailed comparison based on forward modelling of spectral line signatures is still in an exploratory state. That and resolving the apparent discrepancy with regard to the different amounts of overturning motions (§3b(iii)) present in simulations and observations are currently the prime challenges that require further investigation. (c) Modelling of flux emergence and sunspot formation The numerical models described in §3b all have in common that they are initialized with a mostly monolithic sunspot-like field structure and then evolved for a time span of typically a few hours, which is sufficient as far as the development of sunspot fine structure is concerned, but processes that lead to the formation of sunspots in the first place are outside their realm. The formation of sunspots is the part of the larger-scale flux emergence process that transports magnetic fields from within the solar convection zone into the photosphere and beyond into the chromosphere and corona. Currently, there is no single model that can capture the whole process—typically, three regions are modelled independently: (i) the deep convection starting at its base up to about 20 Mm beneath the solar photosphere, (ii) the top of the convection zone starting at about 10–20 Mm depth upwards into the photosphere, and (iii) the outer solar atmosphere starting in the photosphere upwards into the solar corona. Here, we Phil. Trans. R. Soc. A (2012) Downloaded from http://rsta.royalsocietypublishing.org/ on June 18, 2017 Review. Numerical sunspot models 3121 focus entirely on the second region and refer to the review by Fan [3] and the contribution by Archontis [4] for further details regarding modelling the other regions. The flux emergence process in the solar convection zone faces two challenges: the interaction with vigorous convection and the strong density contrast of 106 between the bottom of the convection zone and the photosphere. The strong expansion due to the density drop implies a substantial weakening of magnetic field during the flux emergence process. It was found by Cheung et al. [53] in nearsurface flux emergence simulations that the magnetic field scales with density √ roughly as B ∝ 9, which is primarily a consequence of preferred expansion in the horizontal direction (the expansion along the magnetic field direction does not weaken the field due to the additional stretching term). Assuming that this relation holds over a larger depth range implies that even a field of 100 kG (1 kG = 0.1 T) strength at the base of the convection zone (which is a factor of 10 larger than the equipartition value) is weakened to about 100 G in the photosphere, while a typical sunspot field strength is about 3 kG. Because the pressure scale height decreases from about 50 Mm at the base of the convection zone to about 100 km in the photosphere, most of the density drop is found near the top of the convection zone. As a consequence, a model that focuses on the top 15 Mm of the convection zone includes a density drop of about 104 out of the 106 , while covering only 7.5 per cent of the total convection zone depth. Many effects related to the strong expansion and weakening of rising magnetic fields and the required re-amplification in the photosphere can be studied in numerical models of the near-surface layers. In addition, substantial progress relied heavily on advanced computing resources, as numerical simulations on the scale of the active region (100 Mm) are demanding, even if only a very moderate resolution of about 100 km is adopted. While idealized simulations (many of them neglecting convection) have been used in the past, recently the emergence and interaction of magnetic flux with photospheric granulation was addressed by several groups [54–61]. Flux emergence simulations on the scale of supergranulation and beyond were recently performed by Cheung et al. [53] and Stein et al. [62]. While Stein et al. [62] followed the approach of Stein & Nordlund [54], in which the horizontal field was advected across the bottom boundary condition in upflow regions, Cheung et al. [53] kinematically advected a half torus of flux into the domain, following the approach of Fan & Gibson [63]. The latter approach allows one to introduce a substantially larger amount of flux into the domain, and as a result, Cheung et al. [53] were successful in simulating the formation of a small active region in a 92 × 49 Mm2 wide and 8 Mm deep domain (figure 2). In the early stages, the evolution was dominated by the marked expansion due to the density contrast of about 103 , at later stages the flux was re-amplified in a pair of small sunspots with about 3 × 1021 Mx (108 Mx = 1 Wb) flux, which was about 40 per cent of the flux initially transported into the domain across the bottom boundary. While this simulation showed, for the first time, that the strong weakening of the magnetic field as a consequence of the flux emergence process is consistent with the formation of sunspots at later stages, the overall extent of the domain was not sufficient to clearly quantify the role played by boundary conditions. This work is currently being extended to 150 × 75 Mm2 wide and 16 Mm deep domains. It is found that flux emergence and sunspot formation processes are Phil. Trans. R. Soc. A (2012) Downloaded from http://rsta.royalsocietypublishing.org/ on June 18, 2017 3122 M. Rempel (a) 2 1 Bz (kG) y (Mm) 10 0 –1 log10 |B| (kG) –10 (b) z (Mm) 0 0 –2 –4 –6 –30 –20 –10 0 x (Mm) 10 20 –2 4 2 0 30 Figure 2. The formation of a small sunspot pair with about 3 × 1021 Mx flux. (a) A photospheric magnetogram and (b) the subsurface field strength in the vertical plane running through the simulation domain along the horizontal dotted line in (a). The simulation was performed in a 92 × 49 Mm2 wide and 8 Mm deep domain. This figure is reproduced from Cheung et al. [53] by permission of the American Astronomical Society. (Online version in colour.) only weakly dependent on the initial twist and field strength (10 and 20 kG give essentially similar results) of the rising flux imposed at the bottom boundary. The dynamics are strongly driven by changes in the stratification that are a result of the rising plasma, regardless of field strength. Most importantly, these simulations demonstrate that sunspots can form from fields with about 10 kG field strength (peak value, average value of emerging flux is 5 kG) in 16 Mm depth, which is consistent with the expectations from flux emergence simulations that start at the base of the convection zone with about 50–100 kG field strength. Future improvements will also require an upward extension of the simulation domain into the upper solar atmosphere (chromosphere, corona) to reduce the potential influence from the top boundary condition on the flux emergence process. Overall, these results indicate that a coherent and consistent picture of the flux emergence process in the solar convection zone and photosphere is developing. Over the coming decade, coupled models that describe the whole process from the base of the convection zone into the photosphere and beyond are within reach. 4. Local helioseismology Understanding the subsurface magnetic field and flow structure of sunspots is a central goal of helioseismic studies. Although substantial progress has been made over the past decade (see reviews [5,6,64]), many results remain controversial. Most challenges in applying helioseismic inversion techniques to sunspots have their root in the fact that sunspots cannot be treated as small perturbations—at Phil. Trans. R. Soc. A (2012) Downloaded from http://rsta.royalsocietypublishing.org/ on June 18, 2017 Review. Numerical sunspot models 3123 least not in the upper-most few megametres—which limits the applicability of linear inversion methods. In addition, the effect of the magnetic field cannot be captured by an isotropic wave speed perturbation. Direct applications of time distance helioseismology revealed a two-layer structure underneath sunspots, which is manifest in thermal and velocity structure [64–66]. It was found that the uppermost 4 Mm show a reduction in wave propagation speed and converging flows, while deeper layers show an enhancement of wave speed combined with outflows. The application of ring diagram analysis [67,68] provided results that are comparable on a qualitative level, although the transition from reduced to enhanced wave speed is found in a greater depth of about 7 Mm. It was recently pointed out by Gizon et al. [69,70] and Moradi et al. [71] that many of the observed signatures originate from the upper-most 1 or 2 Mm, where the sunspot imposes a rather large perturbation. Only after properly addressing these near-surface effects are reliable deeper inversions possible. A new promising approach that has been followed recently in parallel by several groups is the forward modelling of wave propagation through stationary sunspot models [72–76]. Using this approach, Cameron et al. [77] demonstrated that the observed helioseismic signatures of the f, p1 and p2 modes can be captured by a rather shallow sunspot model. The strong role of near-surface effects was also deduced by Braun & Birch [78] from the observed frequency dependence of travel time shifts. In addition to the forward modelling of wave propagation through stationary sunspot models, the three-dimensional radiative MHD simulations of sunspots described in §3 can also be used for the study of helioseismic signatures. In these simulations, solar-like oscillations are naturally excited by convection to the extent that the modes are allowed for by the finite size simulation domain. Helioseismic studies require simulations in wider and deeper domains that have a reduced grid resolution to allow for evolution over time scales of at least a few days. Additionally, the larger domain size permits the study of large-scale flow systems such as the moat flow, which was originally found through tracking of magnetic fields surrounding sunspots [79–82] and more recently through helioseismology [69,70,83]. Flow structures on the scale of active regions [84] are currently still out of reach for numerical simulations. With regard to helioseismic signatures, three-dimensional radiative MHD simulations of sunspots also point towards rather shallow wave speed perturbations. The thermal structure (discussed in [71]) as well as large-scale flow structure (figure 3 and [85]) do not show a two-layer structure, as indicated in some of the helioseismic inversions mentioned already. The combination of high-resolution data available today through the SDO, in combination with several forward modelling methods that account for the full complexity of sunspots, makes substantial progress likely in the near future. 5. Summary Over the past decade, substantial progress in our understanding of sunspots resulted from a combination of high-resolution observations and numerical simulations from first principles. The latter benefited substantially from a combination of robust numerical methods with some ‘innovative’ computing Phil. Trans. R. Soc. A (2012) Downloaded from http://rsta.royalsocietypublishing.org/ on June 18, 2017 3124 M. Rempel radial velocity z – zphot (Mm) 0 –2 –4 –6 –8 0 10 30 20 R (Mm) –0.5 0 VR (Vr.m.s.) 0.5 Figure 3. Azimuthally averaged radial flow velocity as function of depth and distance from centre of sunspot. This numerical simulation was evolved for 24 h in a 75 Mm wide and 9 Mm deep computation domain. The velocity amplitude is relative to the convective r.m.s. velocity of undisturbed solar convection at the respective depth level. Positive values indicate outflows. The sunspot is surrounded by an outflow cell at all depth levels. The fast flow concentrated around the indicated t = 1 level is the Evershed flow, which merges in deeper layers with outflows reaching about 50% of the convective r.m.s. velocity over a wide depth range. Further out (R > 22 Mm), a rather superficial photospheric outflow of about 300 m s−1 is present, with indication of an inflow below 4 Mm depth. (Online version in colour.) approaches that resolve the most relevant processes with highest precision, but that also relax other aspects (such as high Alfvén velocities) where possible to keep the computational expense manageable. In addition, the steady increase in computing capabilities allowed these models to pass a threshold at which computations on the scale of entire sunspots with sufficient resolution to capture sunspot fine structure became possible. The simultaneous focus on large and small scales substantially reduces (albeit does not eliminate) uncertainties introduced through initial and boundary conditions and makes the entire simulation more self-consistent with less (hidden) parameters free to adjust. This substantial increase in realism leads to models that simultaneously explain several aspects of sunspot structure, such as magneto-convective energy transport in umbra and penumbra, filamentation and large-scale flow systems without requiring fine tuning, apart from setting up a large-scale magnetic field structure through initial and boundary conditions that resembles a typical sunspot and using a sufficiently high resolution. While these models agree very well with the observationally inferred thermal, magnetic and flow structure of sunspots, a detailed comparison with spectro-polarimetric data in terms of line shapes and polarization signals is still at an exploratory stage. While the currently affordable numerical resolution appears to be sufficient to capture the essence of the underlying magnetoconvection processes, it is still marginal for detailed forward modelling of spectral signatures [47]. Phil. Trans. R. Soc. A (2012) Downloaded from http://rsta.royalsocietypublishing.org/ on June 18, 2017 Review. Numerical sunspot models 3125 On the larger scale, photospheric MHD simulations now approach the scale of active regions and time scales of several days. Work in progress by several groups focuses on the processes involved in the formation and decay of active regions as well as large-scale flow systems, such as the moat flow surrounding sunspots. These simulations provide, as a by-product, very realistic oscillation modes that can be used for an in-depth comparison with helioseismic data from missions such as the SDO or used as a testbed for helioseismic inversion procedures. The latter is, in particular, beneficial for inversions in the proximity of sunspots, which are most relevant for our understanding of solar magnetism. The ultimate goal of studying sun (star) spots is making the connection to the MHD processes in the solar (stellar) interiors responsible for the magnetic field generation in the first place. On the largest scales, the relevant quantities are the overall flux and its distribution over the solar (stellar) surface. This is tied most strongly to the dependence of the underlying dynamo process on rotation rates, stellar luminosities and convection zone depths, and has to be studied in the context of global dynamo models [1]. Details of the surface appearance of spots (including the fine structure) are governed by the properties of the photosphere, most importantly, the gas pressure and Mach number of the photospheric convection. While sunspot models could be generalized to explore a wider (stellar) photospheric parameter range, currently no observational constraints exist for this level of detail. M.R. is partially supported through NASA grant NNH09AK02I (SDO Science Center) to the National Center for Atmospheric Research. The National Center for Atmospheric Research is sponsored by the National Science Foundation. M.R. thanks M. Miesch for very helpful comments on the manuscript. References 1 Miesch, M. 2012 The solar dynamo. Phil. Trans. R. Soc. A 370, 3049–3069. (doi:10.1098/ rsta.2011.0507) 2 Hathaway, D. H. 2010 The solar cycle. Living Rev. Solar Phys. 7, 1. 3 Fan, Y. 2009 Magnetic fields in the solar convection zone. Living Rev. Solar Phys. 6, 4. 4 Archontis, V. 2012 Magnetic flux emergence and associated dynamic phenomena in the Sun. Phil. Trans. R. Soc. A 370, 3088–3113. (doi:10.1098/rsta.2012.0001) 5 Gizon, L. & Birch, A. C. 2005 Local helioseismology. Living Rev. Solar Phys. 2, 6. 6 Gizon, L., Birch, A. C. & Spruit, H. C. 2010 Local helioseismology: three-dimensional imaging of the solar interior. Ann. Rev. Astron. Astrophys. 48, 289–338. (doi:10.1146/annurev-astro082708-101722) 7 Kosovichev, A. G. 2010 Local helioseismology of sunspots: current status and perspectives (invited review). (http://arxiv.org/abs/1010.4927v2) 8 Solanki, S. K. 2003 Sunspots: an overview. Astron. Astrophys. 11, 153–286. (doi:10.1007/s00159003-0018-4) 9 Thomas, J. H. & Weiss, N. O. 2004 Fine structure in sunspots. Ann. Rev. Astron. Astrophys. 42, 517–548. (doi:10.1146/annurev.astro.42.010803.115226) 10 Thomas, J. H. & Weiss, N. O. 2008 Sunspots and starspots. Cambridge, UK: Cambridge University Press. 11 Rempel, M. & Schlichenmaier, R. 2011 Sunspot modeling: from simplified models to radiative MHD simulations. Living Rev. Solar Phys. 8, 3. 12 Borrero, J. M. & Ichimoto, K. 2011 Magnetic structure of sunspots. Living Rev. Solar Phys. 8, 4. 13 Evershed, J. 1909 Radial movement in sun-spots. Mon. Not. R. Astron. Soc. 69, 454. Phil. Trans. R. Soc. A (2012) Downloaded from http://rsta.royalsocietypublishing.org/ on June 18, 2017 3126 M. Rempel 14 Scharmer, G. B., Gudiksen, B. V., Kiselman, D., Löfdahl, M. G. & Rouppe van der Voort, L. H. M. 2002 Dark cores in sunspot penumbral filaments. Nature 420, 151–153. (doi:10.1038/nature01173) 15 Langhans, K., Scharmer, G. B., Kiselman, D., Löfdahl, M. G. & Berger, T. E. 2005 Inclination of magnetic fields and flows in sunspot penumbrae. Astron. Astrophys. 436, 1087–1101. (doi:10.1051/0004-6361:20052678) 16 Márquez, I., Sánchez Almeida, J. & Bonet, J. A. 2006 High-resolution proper motions in a sunspot penumbra. Astrophys. J. 638, 553–563. (doi:10.1086/498668) 17 Rimmele, T. & Marino, J. 2006 The Evershed flow: flow geometry and its temporal evolution. Astrophys. J. 646, 593–604. (doi:10.1086/504794) 18 Scharmer, G. B., Langhans, K., Kiselman, D. & Löfdahl, M. G. 2007 Recent high resolution observations and interpretations of sunspot fine structure. In New solar physics with solar-b mission, vol. 369 (eds K. Shibata, S. Nagata & T. Sakurai), pp. 71–86. Astronomical Society of the Pacific Conference Series. 19 Langhans, K., Scharmer, G. B., Kiselman, D. & Löfdahl, M. G. 2007 Observations of dark-cored filaments in sunspot penumbrae. Astron. Astrophys. 464, 763–774. (doi:10.1051/00046361:20065215) 20 Sánchez Almeida, J., Márquez, I., Bonet, J. A. & Domínguez Cerdeña, I. 2007 The Evershed effect observed with 0.2 angular resolution. Astrophys. J. 658, 1357–1371. (doi:10.1086/511254) 21 Rimmele, T. 2008 On the relation between umbral dots, dark-cored filaments, and light bridges. Astrophys. J. 672, 684–695. (doi:10.1086/523702) 22 Schlichenmaier, R., Rezaei, R., Bello González, N. & Waldmann, T. A. 2010 The formation of a sunspot penumbra. Astron. Astrophys. 512, L1. (doi:10.1051/0004-6361/201014112) 23 Bellot Rubio, L. R., Schlichenmaier, R. & Langhans, K. 2010 Searching for overturning convection in penumbral filaments: slit spectroscopy at 0.2” resolution. Astrophys. J. 725, 11–16. (doi:10.1088/0004-637X/725/1/11) 24 Joshi, J., Pietarila, A., Hirzberger, J., Solanki, S. K., Aznar Cuadrado, R. & Merenda, L. 2011 Convective nature of sunspot penumbral filaments: discovery of downflows in the deep photosphere. Astrophys. J. Lett. 734, L18. (doi:10.1088/2041-8205/734/1/L18) 25 Scharmer, G. B., Henriques, V. M. J., Kiselman, D. & de la Cruz Rodríguez, J. 2011 Detection of convective downflows in a sunspot Penumbra. Science 333, 316–319. (doi:10.1126/science. 1206429) 26 Scharmer, G. B. & Henriques, V. M. J. 2011 SST/CRISP observations of convective flows in a sunspot penumbra. (http://arxiv.org/abs/1109.1301v2) 27 Bharti, L., Joshi, C. & Jaaffrey, S. N. A. 2007 Observations of dark lanes in umbral fine structure from the Hinode solar optical telescope: evidence for magnetoconvection. Astrophys. J. Lett. 669, L57–L60. (doi:10.1086/523352) 28 Tsuneta, S. et al. 2008 The solar optical telescope for the Hinode mission: an overview. Solar Phys. 249, 167–196. (doi:10.1007/s11207-008-9174-z) 29 Ichimoto, K. et al. 2007 Fine-scale structures of the Evershed effect observed by the solar optical telescope aboard Hinode. Publ. Astron. Soc. Jpn. 59, 593. 30 Ichimoto, K. et al. 2007 Twisting motions of sunspot penumbral filaments. Science 318, 1597–1599. (doi:10.1126/science.1146337) 31 Franz, M. & Schlichenmaier, R. 2009 The velocity field of sunspot penumbrae. I. A global view. Astron. Astrophys. 508, 1453–1460. (doi:10.1051/0004-6361/200913074) 32 Bharti, L., Solanki, S. K. & Hirzberger, J. 2010 Evidence for convection in sunspot penumbrae. Astrophys. J. Lett. 722, L194–L198. (doi:10.1088/2041-8205/722/2/L194) 33 Nordlund, A. 1982 Numerical simulations of the solar granulation. I. Basic equations and methods. Astron. Astrophys. 107, 1–10. 34 Nordlund, Å., Stein, R. F. & Asplund, M. 2009 Solar surface convection. Living Rev. Solar Phys. 6, 2. 35 Stein, R. 2012 Magneto-convection. Phil. Trans. R. Soc. A 370, 3070–3087. (doi:10.1098/ rsta.2011.0533) 36 Hurlburt, N. E. & Toomre, J. 1988 Magnetic fields interacting with nonlinear compressible convection. Astrophys. J. 327, 920–932. Phil. Trans. R. Soc. A (2012) Downloaded from http://rsta.royalsocietypublishing.org/ on June 18, 2017 Review. Numerical sunspot models 3127 37 Weiss, N. O., Brownjohn, D. P., Hurlburt, N. E. & Proctor, M. R. E. 1990 Oscillatory convection in sunspot umbrae. Mon. Not. R. Astron. Soc. 245, 434–452. 38 Matthews, P. C., Proctor, M. R. E. & Weiss, N. O. 1995 Compressible magnetoconvection in three dimensions: planforms and nonlinear behaviour. J. Fluid Mech. 305, 281–305. (doi:10.1017/S0022112095004630) 39 Weiss, N. O., Brownjohn, D. P., Matthews, P. C. & Proctor, M. R. E. 1996 Photospheric convection in strong magnetic fields. Mon. Not. R. Astron. Soc. 283, 1153–1164. 40 Hurlburt, N. E., Matthews, P. C. & Proctor, M. R. E. 1996 Nonlinear compressible convection in oblique magnetic fields. Astrophys. J. 457, 933. (doi:10.1086/176786) 41 Hurlburt, N. E., Matthews, P. C. & Rucklidge, A. M. 2000 Solar magnetoconvection. Solar Phys. 192, 109–118. 42 Schüssler, M. & Vögler, A. 2006 Magnetoconvection in a sunspot umbra. Astrophys. J. Lett. 641, L73–L76. (doi:10.1086/503772) 43 Heinemann, T., Nordlund, Å., Scharmer, G. B. & Spruit, H. C. 2007 MHD simulations of penumbra fine structure. Astrophys. J. 669, 1390–1394. (doi:10.1086/520827) 44 Rempel, M., Schüssler, M. & Knölker, M. 2009 Radiative magnetohydrodynamic simulation of sunspot structure. Astrophys. J. 691, 640–649. (doi:10.1088/0004-637X/691/1/640) 45 Rempel, M., Schüssler, M., Cameron, R. H. & Knölker, M. 2009 Penumbral structure and outflows in simulated sunspots. Science 325, 171–174. (doi:10.1126/science.1173798) 46 Rempel, M. 2010 3D numerical MHD modeling of sunspots with radiation transport. In Proc. IAU Symp., Ventura, CA, 22–26 August 2010, vol. 273, p. 6. 47 Remple, M. 2012 Numerical sunspot models: robustness of photospheric velocity and magnetic field structure. Astrophys. J. 750, 62. (doi:10.1088/0004-637X/750/1/62) 48 Bharti, L., Schüssler, M. & Rempel, M. 2011 Can overturning motions in penumbral filaments be detected? Astrophys. J. 739, 35. (doi:10.1088/0004-637X/739/1/35) 49 Zakharov, V., Hirzberger, J., Riethmüller, T. L., Solanki, S. K. & Kobel, P. 2008 Evidence of convective rolls in a sunspot penumbra. Astron. Astrophys. 488, L17–L20. (doi:10.1051/00046361:200810266) 50 Bellot Rubio, L. R., Langhans, K. & Schlichenmaier, R. 2005 Multi-line spectroscopy of dark-cored penumbral filaments. Astron. Astrophys. 443, L7–L10. (doi:10.1051/00046361:200500179) 51 Rempel, M. 2011 Penumbral fine structure and driving mechanisms of large-scale flows in simulated sunspots. Astrophys. J. 729, 5. (doi:10.1088/0004-637X/729/1/5) 52 Scharmer, G. B., Nordlund, Å. & Heinemann, T. 2008 Convection and the origin of Evershed flows in sunspot penumbrae. Astrophys. J. Lett. 677, L149–L152. (doi:10.1086/587982) 53 Cheung, M. C. M., Rempel, M., Title, A. M. & Schüssler, M. 2010 Simulation of the formation of a solar active region. Astrophys. J. 720, 233–244. (doi:10.1088/0004-637X/720/1/233) 54 Stein, R. F. & Nordlund, Å. 2006 Solar small-scale magnetoconvection. Astrophys. J. 642, 1246–1255. (doi:10.1086/501445) 55 Cheung, M. C. M., Schüssler, M. & Moreno-Insertis, F. 2007 Magnetic flux emergence in granular convection: radiative MHD simulations and observational signatures. Astron. Astrophys. 467, 703–719. (doi:10.1051/0004-6361:20077048) 56 Cheung, M. C. M., Schüssler, M., Tarbell, T. D. & Title, A. M. 2008 Solar surface emerging flux regions: a comparative study of radiative MHD modeling and Hinode SOT observations. Astrophys. J. 687, 1373–1387. (doi:10.1086/591245) 57 Abbett, W. P. 2007 The magnetic connection between the convection zone and corona in the quiet sun. Astrophys. J. 665, 1469–1488. (doi:10.1086/519788) 58 Isobe, H., Proctor, M. R. E. & Weiss, N. O. 2008 Convection-driven emergence of small-scale magnetic fields and their role in coronal heating and solar wind acceleration. Astrophys. J. Lett. 679, L57–L60. (doi:10.1086/589150) 59 Martínez-Sykora, J., Hansteen, V. & Carlsson, M. 2008 Twisted flux tube emergence from the convection zone to the corona. Astrophys. J. 679, 871–888. (doi:10.1086/587028) 60 Martínez-Sykora, J., Hansteen, V. & Carlsson, M. 2009 Twisted flux tube emergence from the convection zone to the corona. II. later states. Astrophys. J. 702, 129–140. (doi:10.1088/0004637X/702/1/129) Phil. Trans. R. Soc. A (2012) Downloaded from http://rsta.royalsocietypublishing.org/ on June 18, 2017 3128 M. Rempel 61 Tortosa-Andreu, A. & Moreno-Insertis, F. 2009 Magnetic flux emergence into the solar photosphere and chromosphere. Astron. Astrophys. 507, 949–967. (doi:10.1051/00046361/200912394) 62 Stein, R. F., Lagerfjärd, A., Nordlund, Å. & Georgobiani, D. 2011 Solar flux emergence simulations. Solar Phys. 268, 271–282. (doi:10.1007/s11207-010-9510-y) 63 Fan, Y. & Gibson, S. E. 2003 The emergence of a twisted magnetic flux tube into a preexisting coronal arcade. Astrophys. J. Lett. 589, L105–L108. (doi:10.1086/375834) 64 Kosovichev, A. G. 2006 Subsurface characteristics of sunspots. Adv. Space Res. 38, 876–885. (doi:10.1016/j.asr.2005.04.091) 65 Kosovichev, A. G., Duvall, T. L. J. & Scherrer, P. H. 2000 Time-distance inversion methods and results (invited review). Solar Phys. 192, 159–176. 66 Zhao, J., Kosovichev, A. G. & Duvall, T. L. 2001 Investigation of mass flows beneath a sunspot by time-distance helioseismology. Astrophys. J. 557, 384–388. 67 Basu, S., Antia, H. M. & Bogart, R. S. 2004 Ring-diagram analysis of the structure of solar active regions. Astrophys. J. 610, 1157–1168. 68 Bogart, R. S., Basu, S., Rabello-Soares, M. C. & Antia, H. M. 2008 Probing the subsurface structures of active regions with ring-diagram analysis. Solar Phys. 251, 439–451. (doi:10.1007/ s11207-008-9213-9) 69 Gizon, L. et al. 2009 Helioseismology of sunspots: a case study of NOAA region 9787. Space Sci. Rev. 144, 249–273. (doi:10.1007/s11214-008-9466-5) 70 Gizon, L. et al. 2010 Erratum to: Helioseismology of sunspots: a case study of NOAA region 9787. Space Sci. Rev. 156, 257–258. (doi:10.1007/s11214-010-9688-1) 71 Moradi, H. et al. 2010 Modeling the subsurface structure of sunspots. Solar Phys. 267, 1–62. (doi:10.1007/s11207-010-9630-4) 72 Khomenko, E. & Collados, M. 2006 Numerical modeling of magnetohydrodynamic wave propagation and refraction in sunspots. Astrophys. J. 653, 739–755. (doi:10.1086/507760) 73 Cameron, R., Gizon, L. & Daiffallah, K. 2007 SLiM: a code for the simulation of wave propagation through an inhomogeneous, magnetised solar atmosphere. Astron. Nachr. 328, 313–318. (doi:10.1002/asna.200610736) 74 Parchevsky, K. V. & Kosovichev, A. G. 2007 Three-dimensional numerical simulations of the acoustic wave field in the upper convection zone of the sun. Astrophys. J. 666, 547–558. (doi:10.1086/520108) 75 Hanasoge, S. M. 2008 Seismic halos around active regions: a magnetohydrodynamic theory. Astrophys. J. 680, 1457–1466. (doi:10.1086/587934) 76 Shelyag, S., Zharkov, S., Fedun, V., Erdélyi, R. & Thompson, M. J. 2009 Acoustic wave propagation in the solar sub-photosphere with localised magnetic field concentration: effect of magnetic tension. Astron. Astrophys. 501, 735–743. (doi:10.1051/0004-6361/200911709) 77 Cameron, R. H., Gizon, L., Schunker, H. & Pietarila, A. 2010 Constructing semi-empirical sunspot models for helioseismology. Solar Phys. 268, 293–308. (doi:10.1007/s11207-010-9631-3) 78 Braun, D. C. & Birch, A. C. 2006 Observed frequency variations of solar p-mode travel times as evidence for surface effects in sunspot seismology. Astrophys. J. Lett. 647, L187–L190. (doi:10.1086/507450) 79 Sheeley Jr, N. R. 1969 The evolution of the photospheric network. Solar Phys. 9, 347–357. (doi:10.1007/BF02391657) 80 Sheeley Jr, N. R. 1972 Observations of the horizontal velocity field surrounding sunspots. Solar Phys. 25, 98–103. (doi:10.1007/BF00155747) 81 Harvey, K. & Harvey, J. 1973 Observations of moving magnetic features near sunspots. Solar Phys. 28, 61–71. (doi:10.1007/BF00152912) 82 Brickhouse, N. S. & Labonte, B. J. 1988 Mass and energy flow near sunspots. I. Observations of moat properties. Solar Phys. 115, 43–60. (doi:10.1007/BF00146229) 83 Gizon, L., Duvall Jr, T. L. & Larsen, R. M. 2000 Seismic tomography of the near solar surface. J. Astrophys. Astron. 21, 339. 84 Hindman, B. W., Haber, D. A. & Toomre, J. 2009 Subsurface circulations within active regions. Astrophys. J. 698, 1749–1760. (doi:10.1088/0004-637X/698/2/1749) 85 Rempel, M. 2011 Subsurface magnetic field and flow structure of simulated sunspots. Astrophys. J. 740, 15. (doi:10.1088/0004-637X/740/1/15) Phil. Trans. R. Soc. A (2012)