Magnetoconvection in an inclined magnetic field: linear and weakly non-linear models

Recent observations of sunspots have revealed a rich range of behaviour and a complicated magnetic field structure; magnetoconvection is the key physical process underlying these phenomena. Traditional studies of magnetoconvection have considered vertical, or sometimes horizontal, imposed fields. Tilted fields have received less attention, and yet these are crucial to sunspot dynamics, particularly in the penumbra where field lines are inclined at a variety of angles to the vertical. Tilting the field is also interesting from a purely theoretical viewpoint since it breaks many of the symmetries usually associated with convection problems. In this paper we study the linear stability of a layer permeated by an inclined magnetic field and go on to set up model equations in order to study the patterns formed in the weakly non-linear regime. Possible applications of the results to sunspots are discussed.     

Physical Properties of Wave Motion in Inclined Magnetic Fields within Sunspot Penumbrae

At the surface of the Sun, acoustic waves appear to be affected by the presence of strong magnetic fields in active regions. We explore the possibility that the inclined magnetic field in sunspot penumbrae may convert primarily vertically-propagating acoustic waves into elliptical motion. We use helioseismic holography to measure the modulus and phase of the correlation between incoming acoustic waves and the local surface motion within two sunspots. These correlations are modeled by assuming the surface motion to be elliptical, and we explore the properties of the elliptical motion on the magnetic-field inclination. We also demonstrate that the phase shift of the outward-propagating waves is opposite to the phase shift of the inward-propagating waves in stronger, more vertical fields, but similar to the inward phase shifts in weaker, more-inclined fields.     

Initiation of Coronal Mass Ejections by Sunspot Rotation

We study a filament eruption, two-ribbon flare, and coronal mass ejection (CME) that occurred in NOAA Active Region 10898 on 6 July 2006. The filament was located South of a strong sunspot that dominated the region. In the evolution leading up to the eruption, and for some time after it, a counter-clockwise rotation of the sunspot of about 30 degrees was observed. We suggest that the rotation triggered the eruption by progressively expanding the magnetic field above the filament. To test this scenario, we study the effect of twisting the initially potential field overlying a pre-existing flux-rope, using three-dimensional zero-β MHD simulations. We first consider a relatively simple and symmetric system, and then study a more complex and asymmetric magnetic configuration, whose photospheric-flux distribution and coronal structure are guided by the observations and a potential field extrapolation. In both cases, we find that the twisting leads to the expansion of the overlying field. As a consequence of the progressively reduced magnetic tension, the flux-rope quasi-statically adapts to the changed environmental field, rising slowly. Once the tension is sufficiently reduced, a distinct second phase of evolution occurs where the flux-rope enters an unstable regime characterised by a strong acceleration. Our simulations thus suggest a new mechanism for the triggering of eruptions in the vicinity of rotating sunspots.     

Probing sunspot magnetic fields with p-mode absorption and phase shift data

Long-standing observations of incoming and outgoing f- and p-modes in annuli around sunspots reveal that the spots partially absorb and substantially shift the phase of waves incident upon them. The commonly favoured absorption mechanism is partial conversion to slow magneto-acoustic waves that disappear into the solar interior channelled by the magnetic field of the sunspot. However, up until now, only f-mode absorption could be accounted for quantitatively by this means. Based on vertical magnetic field models, the absorption of p-modes was insufficient. In this paper, we use the new calculations of Crouch & Cally for inclined fields, and a simplified model of the interaction between spot interior and exterior. We find excellent agreement with phase shift data assuming field angles from the vertical in excess of 30° and Alfvén/acoustic equipartition depths of around 600–800 km. The absorption of f-modes produced by such models is considerably larger than is observed, but consistent with numerical simulations. On the other hand, p-mode absorption is generally consistent with observed values, up to some moderate frequency dependent on radial order. Thereafter, it is too large, assuming absorbing regions comparable in size to the inferred phase-shifting region. The excess absorption produced by the models is in stark contrast with previous calculations based on a vertical magnetic field, and is probably due to finite mode lifetimes and excess emission in acoustic glories. The excellent agreement of phase shift predictions with observational data allows some degree of probing of subsurface field strengths, and opens up the possibility of more accurate inversions using improved models. Most importantly, though, we have confirmed that slow mode conversion is a viable, and indeed the likely, cause of the observed absorption and phase shifts.     

Magnetic and thermal phase shifts in the local helioseismology of sunspots

Phase perturbations due to inclined surface magnetic field of active region strength are calculated numerically in quiet Sun and simple sunspot models in order to estimate and compare the direct and indirect (thermal) effects of the fields on helioseismic waves. It is found that the largest direct effects occur in highly inclined field characteristic of penumbrae, and scale roughly linearly with magnetic field strength. The combined effects of sunspot magnetic and thermal anomalies typically yield negative travel-time perturbations in penumbrae. Travel-time shifts in umbrae depend on details of how the thermal and density structure differs from the quiet Sun. The combined shifts are generally not well approximated by the sum of the thermal and magnetic effects applied separately, except at low field strengths of around 1 kG or less, or if the thermal shift is small. A useful rule-of-thumb appears to be that travel-time perturbations in umbrae are predominantly thermal, whereas in penumbrae they are mostly magnetic.     

Vector magnetogram and dopplergram observation of magnetic flux emergence and its explanation

During 23–28 August 1988, at the Huairou Solar Observation Station of Beijing Observatory, the full development process of the region HR 88059 was observed. It emerged near the center of the solar disk and formed a medium active region. A complete series of vector magnetograms and photospheric and chromospheric Dopplergrams was obtained. From an analysis of these data, combined with some numerical simulations, the following conclusions can be drawn. (1) The emergence of new magnetic flux from enhanced networks followed by sunspot formation is an interesting physical process which can be simply described by MHD numerical simulation. The phenomena accompanying it occur according to a definite law summarized by Zwaan (1985). The condition for gas cooling and sunspot formation seems to be transverse field strength > 50 G together with longitudinal field strength > 700 G. For a period of 4 to 5 hours, the orientation of the transverse field shows little change. The configuration of field lines may be derived from vector magnetograms. The arch filament system can be recognized as an MHD shock. (2) New opposite bipolar features emerge within the former bipolar field with an identical strength which will develop a sunspot group complex. Also, arch filament systems appear there located in the position of flux emergence. The neutral line is often pushed aside and curved, leading to faculae heating and the formation of a current sheet. In spite of complicated Dopplergrams, the same phenomena occur at the site of flux emergence as usual: upward flow appears at the location of the emerging and rapidly varying flux near the magnetic neutral line, and downdraft occurs over large parts of the legs of the emerging flux tubes. The age of magnetic emerging flux (or a sunspot) can be estimated in terms of transverse field strengths: when 50 G < transverse field < 200 G, the longitudinal magnetogram and Dopplergram change rapidly, which indicates a rigourously emerging magnetic flux. When the transverse field is between 200 and 400 G, the area concerned is in middle age, and some of the new flux is still emerging there. When the transverse field > 400 G, the variation of the longitudinal magnetogram slows down and the emerging arch becomes relatively stable and a photospheric Evershed flow forms at the penumbra of the sunspot.     

Analysis of the Helioseismic Power-Spectrum Diagram of a Sunspot

The continuous high spatial resolution Doppler observation of the Sun by the Solar Dynamics Observatory/Helioseismic and Magnetic Imager allows us to compute a helioseismic k–ω power-spectrum diagram using only oscillations inside a sunspot. Individual modal ridges can be clearly seen with reduced power in the k–ω diagram that is constructed from a 40-hour observation of a stable and round sunspot. Comparing this with the k–ω diagram obtained from a quiet-Sun region, one sees that inside the sunspot the f-mode ridge is more reduced in power than the p-mode ridges, especially at high wavenumbers. The p-mode ridges all shift toward lower wavenumber (or higher frequency) for a given frequency (or wavenumber), implying an increase of phase velocity beneath the sunspot. This is probably because the acoustic waves travel across the inclined magnetic field of the sunspot penumbra. Line-profile asymmetries exhibited in the p-mode ridges are more significant in the sunspot than in the quiet Sun. Convection inside the sunspot is also highly suppressed, and its characteristic spatial scale is substantially larger than the typical convection scale of the quiet Sun. These observational facts demand a better understanding of magnetoconvection and interactions of helioseismic waves with magnetic field.     

Helioseismology of Sunspots: Confronting Observations with Three-Dimensional MHD Simulations of Wave Propagation

The propagation of solar waves through the sunspot of AR?9787 is observed by using temporal cross-correlations of SOHO/MDI Dopplergrams. We then use three-dimensional MHD numerical simulations to compute the propagation of wave packets through self-similar magnetohydrostatic sunspot models. The simulations are set up in such a way as to allow a comparison with observed cross-covariances (except in the immediate vicinity of the sunspot). We find that the simulation and the f-mode observations are in good agreement when the model sunspot has a peak field strength of 3 kG at the photosphere and less so for lower field strengths. Constraining the sunspot model with helioseismology is only possible because the direct effect of the magnetic field on the waves has been fully taken into account. Our work shows that the full-waveform modeling of sunspots is feasible.     

Fractal properties of solar magnetic fields

We study the spatial properties of solar magnetic fields using data from the Solar Vector Magnetograph of the Marshall Space Flight Center (MSFC) (FeI 5250.2 Å) and SOHO/MDI longitudinal magnetic field measurements (Ni 6767.8 Å) (96-min full-disk maps). Our study is focused on two objects: the fractal properties of sunspots and the fractal properties of the spatial magnetic field distribution of active and quiet regions considered as global structures. To study the spatial structure of sunspots, we use a well-known method of determining the fractal dimension based on an analysis of the perimeter—area relation. To analyze the fractal properties of the spatial magnetic field distribution over the solar surface, we use a technique developed by Higuchi. We have revealed the existence of three families of self-similar contours corresponding to the sunspot umbra, penumbra, and adjacent photosphere. The fractal coefficient has maxima near the umbra—penumbra and penumbra—photosphere boundaries. The fractal dependences of the longitudinal and transverse magnetic field distributions are similar, but the fractal numbers themselves for the transverse fields are larger than those for the longitudinal fields approximately by a factor of 1.5. The fractal numbers decrease with increasing mean magnetic field strength, implying that the magnetic field distribution is more regular in active regions.     

Common Characteristics of the Active Regions of Strong Proton Flares

Common characteristics of nine active regions with strong proton flares in the 22nd solar activity cycle have been presented. Results show that the typical morphology of these active regions is a -type sunspot with a single multiple structure, in which there are many umbras with different magnetic polarities, packed tightly by a single penumbra. In these active regions, the rotating directions of the sunspot groups are nearly independent of their position on the solar disk. When the angle of rotation approaches the positive or the negative maximum, proton flares may occur in these active regions. After proton flares, sunspot groups rotate in the inverse direction because of the slack in the flux rope.     

Basic properties of sunspots: equilibrium,stability and long-term eigen oscillations

A number of fundamental questions as regards the physical nature of sunspots are formulated. In order to answer these questions, we apply the model of a round-shaped unipolar sunspot with a lower boundary consisting of cool plasma and with strong magnetic field at the depth of about 4 Mm beneath the photosphere, in accordance with the data of local helioseismology and with certain physically sound arguments (the shallow sunspot model). The magnetic configuration of a sunspot is assumed to be close to the observed one and similar to the magnetic field of a round solenoid of the appropriate size. The transverse (horizontal) and longitudinal (vertical) equilibria of a sunspot were calculated based on the thermodynamic approach and taking into account the magnetic, gravitational, and thermal energy of the spot and the pressure of the environment. The dependence of the magnetic field strength in the sunspot center, B ₀, on the radius of the sunspot umbra a is derived theoretically for the first time in the history of sunspot studies. It shows that the magnetic field strength in small spots is about 700 Gauss (G) and then increases monotonically with a, tending asymptotically to a limit value of about 4000 G. This dependence, B ₀(a) includes, as parameters, the gravity acceleration on the solar surface, the density of gas in the photosphere, and the ratio of the radius of the spot (including penumbra), a _p, to the radius of its umbra a. It is shown that large-scale subsurface flows of gas in the sunspot vicinity, being the consequence but not the cause of sunspot formation, are too weak to contribute significantly to the pressure balance of the sunspot. Stability of the sunspot is provided by cooling of the sunspot plasma and decreasing of its gravitational energy due to the vertical redistribution of the gas density when the geometric Wilson depression of the sunspot is formed. The depth of a depression grows linearly with B ₀, in contrast to the quadratic law for the magnetic energy. Therefore, the range of stable equilibria turns out to be limited: large spots, with radius a larger than some limit value (about 12–18 Mm, depending on the magnetic field configuration), are unstable. It explains the absence of very large spots on the Sun and the appearance of light bridges in big spots that divide the spot into a few parts. The sunspots with B ₀≈2.6÷2.7 kilogauss (kG) and a≈5 Mm are most stable. For these spots, taken as a single magnetic structure, the period of their vertical eigen oscillations is minimal and amounts, according to the model, to 10–12 hours. It corresponds well to the period derived from the study of long-term oscillations of sunspots using SOHO/MDI data.     

The role of emerging bipoles in the formation of a sunspot penumbra

The generation of magnetic flux in the solar interior and its transport from the convection zone into the photosphere, the chromosphere, and the corona will be in the focus of solar physics research for the next decades. With 4 m class telescopes, one plans to measure essential processes of radiative magneto‐hydrodynamics that are needed to understand the nature of solar magnetic fields. One key‐ingredient to understand the behavior of solar magnetic field is the process of flux emergence into the solar photosphere, and how the magnetic flux reorganizes to form the magnetic phenomena of active regions like sunspots and pores. Here, we present a spectropolarimetric and imaging data set from a region of emerging magnetic flux, in which a proto‐spot without penumbra forms a penumbra. During the formation of the penumbra the area and the magnetic flux of the spot increases. First results of our data analysis demonstrate that the additional magnetic flux, which contributes to the increasing area of the penumbra, is supplied by the region of emerging magnetic flux. We observe emerging bipoles that are aligned such that the spot polarity is closer to the spot. As an emerging bipole separates, the pole of the spot polarity migrates towards the spot, and finally merges with it. We speculate that this is a fundamental process, which makes the sunspot accumulate magnetic flux. As more and more flux is accumulated a penumbra forms and transforms the proto‐spot into a full‐fledged sunspot (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Solar Magnetograms at 12 μm Using the Celeste Spectrograph

We present the first solar vector magnetogram constructed from measurements of infra-red Mg I 12.32-μm line spectra. Observations were made at the McMath-Pierce Telescope using the Celeste spectrometer/polarimeter. Zeeman-split Stokes line spectra were fitted with Seares profiles to obtain the magnetic field parameters. Maps of absolute field strength, line-of-sight angle, and azimuth are presented. Analysis shows that the variation in field strength within a spatial resolution element, 2 arcseconds, is greatest in the sunspot penumbra and that this is most likely caused by vertical field strength gradients, rather than horizontal image smearing. Widths of the Zeeman-split σ components, assuming a formation layer thickness of 200 km, indicate that vertical field strength gradients can be as large as 6.5 G/km in a penumbra.     

Two-Dimensional Spectroscopy of Photospheric Shear Flows in a Small <Emphasis Type="Italic">δ</Emphasis> Spot

In recent high-resolution observations of complex active regions, long-lasting and well-defined regions of strong flows were identified in major flares and associated with bright kernels of visible, near-infrared, and X-ray radiation. These flows, which occurred in the proximity of the magnetic neutral line, significantly contributed to the generation of magnetic shear. Signatures of these shear flows are strongly curved penumbral filaments, which are almost tangential to sunspot umbrae rather than exhibiting the typical radial filamentary structure. Solar active region NOAA 10756 was a moderately complex β δ sunspot group, which provided an opportunity to extend previous studies of such shear flows to quieter settings. We conclude that shear flows are a common phenomenon in complex active regions and δ spots. However, they are not necessarily a prerequisite condition for flaring. Indeed, in the present observations, the photospheric shear flows along the magnetic neutral line are not related to any change of the local magnetic shear. We present high-resolution observations of NOAA 10756 obtained with the 65-cm vacuum reflector at Big Bear Solar Observatory (BBSO). Time series of speckle-reconstructed white-light images and two-dimensional spectroscopic data were combined to study the temporal evolution of the three-dimensional vector flow field in the β δ sunspot group. An hour-long data set of consistent high quality was obtained, which had a cadence of better than 30 seconds and subarcsecond spatial resolution.     

Oscillatory motions in sunspots

We observe vertical velocity oscillations in some sunspot umbrae with periods of about 180 s and peak to peak amplitudes up to 1 km s^–1. These oscillations are not visible in either the line depth, line width or the continuum intensity. No correlation seems to exist between the occurence of these oscillations and the presence of the chromospheric umbral flashes (Solar Phys. 7, 351, 1069). In the spot penumbra there is an indication of a long period oscillation, the period increasing from about 300 s in the inner penumbra to nearly 1000 s at the penumbra-photosphere boundary. An attempt has been made to interpret these oscillations in terms of gravity or acoustic waves, travelling along the magnetic field lines, taking into account the variation of scale height and magnetic field direction across the sunspot.     

On the outer boundary of the sunspot penumbra

Comparison of photographic observations and vector-magnetograph measurements demonstrate that the outer boundary of the sunspot penumbra – even in complex sunspot groups – closely follows the 0.075 T iso-gauss line of the total value of the magnetic field, corresponding approximately to the equipartition value in the photosphere. Radio observations also show this feature. The thick penumbra model with interchange convection (Jahn and Schmidt, 1994) gives the best explanation of the penumbral structure. Supplementary material to this paper is available in electronic form at http://dx.doi.org/10.1023/A:1020985530075     

MHD simulations of sunspot rotation and the coronal consequences

A simplified magnetic configuration is used to model some aspects of observations of a rotating sunspot and its overlying coronal loops. In the observations a large sunspot rotates over a few days and two smaller pores spiral into it. The coronal loops become sigmoidal in shape and flares are seen in Yohkoh/SXT and GOES. We have modeled the sunspot, one of the pores and the loops connecting these to a diffuse region of plasma of the opposite polarity. Two sets of MHD simulations are considered: (i) rotation of the sunspot and pore alone and (ii) rotation of the sunspot with inflow of the pore. Rotation alone can trigger the ideal kink instability in the loops but only for a rotation that is much greater than the observed value. There is no build-up of current which is needed for magnetic reconnection to occur. However, when inflow is included a strong build-up of current is seen as the pore merges with the sunspot. Comparing these results from the simulations with the observations, we find that the observed merging of the pores coincides with the timing of the flare. Therefore, we suggest that the merging of the pores with the large sunspot may be responsible for the flaring.     

Development of a Code to Analyze the Solar White-Light Images from the Kodaikanal Observatory: Detection of Sunspots,Computation of Heliographic Coordinates and Area

Sunspots are the most conspicuous aspects of the Sun. They have a lower temperature, as compared to the surrounding photosphere; hence, sunspots appear as dark regions on a brighter background. Sunspots cyclically appear and disappear with a 11-year periodicity and are associated with a strong magnetic field ( ～10³ G) structure. Sunspots consist of a dark umbra, surrounded by a lighter penumbra. Study of umbra–penumbra area ratio can be used to give a rough idea as to how the convective energy of the Sun is transported from the interior, as the sunspot’s thermal structure is related to this convective medium.An algorithm to extract sunspots from the white-light solar images obtained from the Kodaikanal Observatory is proposed. This algorithm computes the radius and center of the solar disk uniquely and removes the limb darkening from the image. It also separates the umbra and computes the position as well as the area of the sunspots. The estimated results are compared with the Debrecen photoheliographic results. It is shown that both area and position measurements are in quite good agreement.     

Magnetic field inclination and atmospheric oscillations above solar active regions

Recent observational evidence for magnetic field direction effects on helioseismic signals in sunspot penumbrae is suggestive of magnetohydrodynamic (MHD) mode conversion occurring at lower levels. This possibility is explored using wave mechanical and ray theory in a model of the Sun's surface layers permeated by uniform inclined magnetic field. It is found that fast-to-slow conversion near the equipartition depth at which the sound and Alfvén speeds coincide can indeed greatly enhance the atmospheric acoustic signal at heights observed by Solar and Heliospheric Observatory/Michelson Doppler Imager and other helioseismic instruments, but that this effect depends crucially on the wave attack angle , i.e. the angle between the wavevector and the magnetic field at the conversion/transmission depth. A major consequence of this insight is that the magnetic field acts as a filter, preferentially allowing through acoustic signal from a narrow range of incident directions. This is potentially testable by observation.     

Equilibrium and decay of sunspots

We suggest a quantitative sunspot model developed in terms of mean-field magnetohydrodynamics (MHD). The model consistently describes the distributions of magnetic field, fluid velocity, and thermodynamic parameters in a sunspot and the surrounding matter. Two versions of the model allow the MHD equilibrium in sunspots and their slow decay to be analyzed. The baroclinic flow converging to the sunspot plays an important role in the equilibrium. Several calculated characteristics—almost uniform distributions of brightness and magnetic field inside sunspots, their abrupt changes at the boundary, and nearly linear decreases in the area and magnetic flux of decaying sunspots with time—qualitatively agree with the observations.