Solar MHD Theory and Observations: A High Spatial Resolution Perspective ASP Conference Series, Vol. 354, 2006 Uitenbroek, Leibacher, and Stein

Connections: Photosphere – Chromosphere - Corona

Boris V. Gudiksen Institute for Theoretical Astrophysics, University of Oslo, Oslo, Norway

Abstract. The chromosphere is not only the region where the atmosphere goes from being optically thick to optically thin, but also the region where the dynamics changes from being controlled by the plasma to being controlled by the magnetic field. The magnetic field changes from being concentrated in small regions to being space filling. This expansion has traditionally been modeled by the magnetic funnel or wine-glass picture. For several reasons it is hard to gain any information about the magnetic field in this region, so this model remains unconfirmed. Three recent methods to acquire magnetic field information from this interesting region will be reviewed, and I will argue that the results from such investigations will require that we heavily revise the simplistic magnetic funnel picture.

1. Introduction

The solar atmosphere from the photosphere to the corona, is subject to large changes in the magnetic structure. At chromospheric heights the magnetic field fans out and becomes volume-filling. The magnetic connection between the photosphere and the corona depends both on the magnetic field structure in the photosphere as well as the dynamics of the upper chromosphere, making this problem highly complicated. Because the magnetic field confines plasma along the magnetic field lines, the magnetic field geometry becomes vital for the interpretation of observational data. The rapid expansion of the magnetic field from the photosphere to the low corona is due to the very small pressure scale height in the photosphere and chromosphere, which makes β (= Pgas/PB) change from large values in most of the photosphere (except e.g. in sunspots) to small values in the corona. (β is a measure of the importance of the magnetic pressure relative to the gas pressure in controlling the dynamics of the plasma.) This expansion is also apparent in observations of the solar atmosphere in, for instance, Ca ii H or K line and Hα. The formation of these lines, especially Hα, is complicated and stretches over a range in heights and there is a striking difference between them. Images in filters isolating the Ca ii lines are formed in the lower chromosphere and show no major difference in morphology from images showing photospheric lines. Granulation, even though it might be inverted, is still visible, and magnetic concentrations are lit up in these lines, but there is no imprint of a space filling magnetic field. Images in Hα, on the other hand, formed in the upper chromosphere are heavily influenced by a space filling magnetic field. The limb effect on certain spectral lines also requires a rapid expansion of the magnetic field with increasing height. The conclusion therefore must be that the magnetic field goes from being concentrated in small regions in the lower chromosphere expanding over one Mm or less to be completely space filling. The evidence for the fast transition of the magnetic field from being highly confined in the lower chromosphere to being space filling in the upper chromosphere led Giovanelli (1980) to produce a picture of the field that has since been referred to as funnels or “wine-glass” configuration. The idea was that the magnetic field concentrations in the magnetic network consists of near vertical field, that when reaching the upper chromo- sphere turns almost horizontal, before it turns vertical again when meeting flux from 331 332 Gudiksen

Figure 1. Loop model created by Peter (2001) to explain the increase in DEM for lower temperatures a another flux concentration. It was assumed that the network was the foot points of these funnels, and their width were comparable to the typical dimension of the network, i.e. of super granular size. This idea has stuck and has been accepted even though there has been no actual measurement of the magnetic field having this wine-glass shape over super granular scales. Only recently has it actually been possible to measure the magnetic field in the chromosphere at a resolution and time cadence that was usable (Solanki et al. 2003), even though we would still like to have better cadence and much better resolution. Apart from this there is very little observational and theoretical work on the magnetic connection between the photosphere, the chromosphere and the corona. In this review I will try to give a overview of what we know from the work that has been done on this subject and what is being done to get to the bottom of this problem. I will also give a biased view of what the magnetic field might look like, especially in the interface region around the important β = 1 surface.

2. The Funnel

As already mentioned, the funnel model came into life in 1980 when Giovanelli tried to explain the change of transition region lines from disk center to the limb. The model in its simplest form has only one free parameter, the expansion factor. This factor describes the relative expansion from the photosphere to the corona. The simplicity of the model, and the lack of knowledge of the magnetic field configuration around the β = 1 surface has made this model the preferred one for many years. Lately, a number of observations and theoretical work have put this model under considerable strain. One problem came with the observation of increasing differential emission measure (DEM) at low temperatures, which the funnel picture had a hard time explaining. In order to resolve the DEM problem, the funnel picture was expanded with a number of low lying short loops filled with cool material in order to increase the amount of emitting material at chromospheric temperatures (see Figure 1). This makes the funnel picture lose one of its main advantages – its simplicity. This picture is most likely also problematic due to the fact that this layer in the solar atmo- sphere is at β ≥ 1. That makes it possible for the dynamics to drive reconnection, and make the short loops reconnect with the overlaying funnels. The effect would destroy Connections: Photosphere – Chromosphere – Corona 333 the nice wine-glass configuration, and so the funnel picture with the extension of the low-lying loops seems untenable.

3. Getting the Magnetic Configuration

There are at least three main approaches one can follow in order to get information about the magnetic configuration in the solar atmosphere. All three have strengths and weaknesses, and I will try to highlight some of them here. All of them have only been possible with the equipment that has come online within the last few years, so it would be unfair to judge them at this early time in their development.

3.1. Direct Observations The direct observations of the magnetic field above the photosphere is complicated for a number of reasons. Traditionally the Zeeman effect which is a splitting of magnetically sensitive lines is used. However, in order to get a signal it is preferable to have a line with a high Land´e factor. It cannot be chosen just on the basis of the Land´e factor, because the magnetic shift of the line not only depends on the Land´e factor but also on wavelength of the line. This makes long wavelengths easier to observe than shorter wavelengths. The wavelength, of course, also decides the theoretical spatial resolution, and furthermore one has to find a strong line in order to observe the line at all. This narrows the number of lines to just a few. The aim is to get all three components of the magnetic field vector, but that involves four Stokes parameters, I (the line intensity), V (circular polarization), U and Q (linear polarization oriented at a 90◦angle with respect to one another). U and Q are usually on the order of just 1% of I, making the perpendicular components of the magnetic field more susceptible to noise and one needs relatively long exposure times to get good statistics. In order to do a reasonable inversion of the signal, the line has to be resolved either by using a spectrograph and rastering to build up an image, or by imaging in very narrow wavelength windows, sampling the line profile at a number of wavelengths. Since both seeing and the Sun change on a time scale smaller or comparable to the time necessary to build up a data set, this creates problems when inverting the signals obtained. For small values of the magnetic field strength, the V signal is roughly proportional to the magnetic field strength, but the signal gets saturated at larger values of the field strength, in principle, setting only a lower limit for the field strength and it provides only the line of sight component of the magnetic field. A complementary method is to use the Hanle effect. The Hanle effect surfaces when a radiation field passes an inclined magnetic field, effectively creating a difference in populations of otherwise magnetically degenerate atomic levels. It has the opposite problem of the Zeeman effect when extracting the magnetic field strength, since the Hanle effect is sensitive mainly to low field strengths, it effectively sets an upper limit to the field strength. Furthermore to observe the Hanle effect, lines formed rather high in the solar atmosphere are needed, because it depends on a scattering process that is erased by the dense lower atmosphere. The Hanle effect can also provide data on opposite polarities within a resolution element, which cancels out for the Stokes signal, providing a good tool for studying low flux concentrations. In spite of these problems, it was recently possible for Solanki et al. (2003) to recreate the 3D magnetic field by observing the chromospheric He i triplet at 1083 nm and the Si i line at 1082.7 nm, to get two vector magnetograms at two different heights. Even though this is quite an accomplishment, it might not have produced a too surprising result. The resolution of the observations were 1′′. 5, and scanning through the image and the wavelengths needed, took roughly 12 minutes. Because of this long integration time, most of the small scale signal will naturally be smeared out leaving only large scale structures. From observations of the photospheric vector magnetogram, and a following non-linear force-free extrapolation, it is often possible to reproduce the large 334 Gudiksen scale structure of the magnetic field as deduced from EUV imaging of coronal loops. It is therefore questionable if the procedure employed by Solanki et al. (2003) has any practical use, until the integration time can be reduced to at least below the granular time scale of roughly five minutes. 3.2. Extrapolation Procedures There exists a number of methods for extrapolating a full 3D coronal magnetic field from a vector magnetic field in the photosphere. All of them assume that the field in the layer between the photosphere and the corona does not change characteristics, i.e. that the whole of the atmosphere has β ≪ 1. Because of the observational difficulties of observing the magnetic field in the chromosphere and the transition region, we do not know how big an effect this erroneous assumption has on the coronal part of the field. We can say that in general the magnetic field in the chromosphere and transition region cannot be reproduced by this method. The extrapolated field can be in three different forms all of which produce no Lorenz force, and so are solutions of

∇ × B = α(x) B (1) ∇ · B = 0 (2) where B is the magnetic field and the left hand side of equation (1) is proportional to the electric current. α is a parameter that, depending on its value and spatial distribution, decides on the three general types of force free fields. If α is zero everywhere, then the field is potential, and one can show that this is the minimum energy the field can have under the chosen boundary conditions. Giving α a non-zero but constant value for all x produces a linear force free field, while having α vary as a function of position produces a non-linear force free field. By taking the divergence of equation (1) on gets B · ∇α = 0 and so α must be constant along magnetic field lines, making it necessary to solve for B taking α into account, while α is unknown in space until B is known. That is what makes the problem very non-linear and mathematically hard to solve and in general it can be hard to show that a solution exists for a given boundary condition. Take for instance the case where α is given at the lower boundary in a box, but that not all field lines penetrate the lower boundary. That would make the number of solutions infinite since α on the field lines not penetrating the lower boundary is undetermined. In general though, this is not a problem, since for the Sun the magnetic field is rooted in the photosphere. Three main methods have been used to solve for the non-linear force free case the last few years. Before these methods, extrapolations were in general produced by computa- tionally expensive finite difference schemes, that solved the equations by brute force. The first method was developed and implemented by Yan & Sakurai (1997, 2000) and uses a Green’s function like method where a reference function is invoked in order to turn a volume integral over B and the simple reference function Y into a surface integral around a small volume, in which B can now be computed. This method is best suited for open boundaries, where one can assume that the B decreases fast towards the outer boundary. Another method developed by Wheatland et al. (2000) and implemented, for instance, by Wiegelmann (2004) and McTiernan et al. (2004) ( called the optimization method ) involves an evolution of the magnetic field through minimization of a function that de- pends on how well the magnetic field satisfies Equations ( 1 ) and (2 ). One calculates a change for the magnetic field, that will minimize the difference function, eventually reaching a state where the magnetic field is solonoidal and satisfies the boundary con- ditions to a predetermined accuracy. The last method mentioned here was developed by Grad & Rubin (1958) and im- plemented independently by R´egnier et al. (2002); R´egnier & Amari (2004) based on the work by Amari et al. (1997, 1999) and by Wheatland (2004). These methods are Connections: Photosphere – Chromosphere – Corona 335 iterative and are implemented slightly differently by the two groups. But both of these methods uses only data on α in one polarity, since the return flux from one polarity must have the same distribution of α at each polarity, and in general observations can- not satisfy that because of the observational problems, and because of field of view effects. If not all of the active region is included, or just the surrounding network, then it is impossible to include only the flux concentrations that will make the distribution of α the same in each polarity. Since these procedures are iterative, they would never reach a solution if the distributions were not alike, and so both of these methods ignore the values of α in one polarity in order to get around this problem. R´egnier et al. use a potential field extrapolation as initial condition, and then injects current in the mid- dle of the magnetic field. Wheatland uses the current distribution in one polarity and evolve the a number of chosen field lines containing the chosen current by using an exact solution to Ampere’s law. Which of the three methods produces the best results is at this point unknown but with the present implementations, the Wheatland Grad-Rubin method seems to be significantly faster than the rest. The question of how well these methods reproduce the coronal field is a question of some importance, but giving a complete answer is not easy as the next section will explain.

3.3. Guided Extrapolation Procedures

This is a method that builds on a composite approach. It is being investigated by the NLFFF workgroup at LMSAL. The idea is to use the extrapolation procedures and guide them to a solution that also agrees with EUV observations of the solar corona by space craft as SOHO, TRACE, and especially Solar-B. There are several problems to solve before this will be possible. Primarily we don’t know how well the 2D observational coronal data will translate into conditions for the extrapolations to solve. For the methods presented here, one would have to identify a field line in the 3D extrapolation that would best represent a loop in the observations, and then change α to get a better fit, but changing α even along only a small part of the total flux in the photosphere could in principle change the whole structure of the magnetic field in the corona. This could create a situation where the field line-loop pair lies far apart after the change of α. Creating an implementation that can handle such cases seems at this point hard, but it depends on how often such a large reordering of the field happens when using observational data.

4. Outer Corona & The Interplanetary Field

The outer corona, here defined as the part of the corona that is at the interface with the solar wind and not affected very much by the active regions, can be modeled very well by the flux surface model. The flux surface model is simply a potential extrapolation of observed magnetic field in the photosphere with the boundary condition that it be radial at the “flux surface”. This surface is generally placed at different radii, but usually around 2.5 R⊙. This produces a field that agrees well with the magnetic field inferred from observations. Outside this radii the Parker solar wind spiral (Parker 1958) takes over. The field is slightly more complicated than the original picture presented by Parker since the velocity, assumed constant by Parker, is different along different field lines, so the field lines coming from different parts of the corona will not have the same curvature, and thus interact so the field will be more complicated but in general agree well with the Parker spiral. 336 Gudiksen

Figure 2. 3D simulation of the solar corona readily shows loops in simulated TRACE 171 A(top˚ left) and TRACE 195 A(top˚ right), emanating from the photospheric magnetic field (bottom). The periodic simulation box is shown three times side by side in this figure

Dynamics and Modeling

The solar atmosphere at and below the β = 1 surface is known to be highly dynamic. The magnetic field in this region must consequently be quite jumbled. In the highly dynamic atmosphere, the field will be forced to reconnect as velocities will drive field lines together. Depending on the local value of β this process may be efficient or less so, but it will make the field relax towards a lower energy. We cannot say how efficient this process is in relaxing the field, because we don’t know the reconnection speed or the efficiency of the velocity field to create magnetic tangential discontinuities. Most likely the jumbling of the field lines in the layer between the photosphere and the β = 1 surface will on average cancel out if one only looks at granular scales so that the extrapolations from a photospheric vector magnetogram can be used as input to get the coronal field (Gudiksen & Nordlund 2005a). The questions is what happens when there are large scale velocity fields present or if the field is already twisted at emergence. Since the extrapolations are generally used to examine field configurations before coronal events such as flares or coronal mass ejections, this question is highly relevant. The answer will probably depend on scale. The solar photospheric velocity field follows power laws all the way from granular scales to super granular scales, and maybe beyond. One relation is the velocity amplitude that goes as k−1, where k is the spatial wavenumber. Similarly the lifetime of the granular structures goes as k2. These relations most likely means that the field, on a certain scale, will not deviate from its original structure, until at least a turn over time at that scale has passed. Since now the original structure is probably not potential, one could argue that as long as a few turn-over times have passed, the field will be relaxed on that scale. This might be a too severe simplification. An example would be a highly twisted field, that near one foot point is almost horizontal due to its internal twist. In order to relax, there would have to be enough flux, to which the twist could be transferred in order to relax to a low energy state which would not be the case if, for instance, the twisted field were the flux in the main sunspot of an active region. Connections: Photosphere – Chromosphere – Corona 337

Figure 3. Loops originating in the magnetogram (black line, left) from the simulation show no resemblance to the traditional funnel picture (right)

The details of the magnetic field structure in the chromosphere is still unknown, but several hints have surfaced lately that makes the funnel picture in its simplest form problematic. A number of observations have shown that the inter network is not field free (this has been known for a long time), but has a large amount of flux hiding in low flux concentrations. How exactly this flux is distributed in the photosphere and how much is present is still being discussed (see for instance Dom´ınguez Cerdena˜ et al. 2003). The effect of the low flux concentrations were highlighted by Schrijver & Title (2003), who showed that for a network concentration surrounded by low flux concentrations consistent with observations, the magnetic field looked nothing like a wine-glass. It has to be remarked that Schrijver & Title (2003) used a potential field extrapolation method to get the 3D magnetic field, and this in itself makes the field from one polarity reach for the nearest opposite polarity. Even though the field in the chromosphere is most likely not potential because of the vigorous motions in the photosphere and chromosphere, these same motions drive reconnection that relaxes the field on a time scale that can be sufficiently small, making a picture like Figure 1 unrealistic. Another piece of work that talks against the wine-glass picture is by Gudiksen & Nordlund (2005b,a) that did a 3D MHD simulation of the solar atmosphere from the photosphere to 37 Mm up into the corona. The magnetic field was extrapolated from an observation by SOHO- MDI of AR 9114 in order to get a typical active region magnetic flux distribution in the photosphere. The magnetogram was extrapolated into a potential field, and then driven in the photosphere by a driver that closely resembles the photospheric velocity field geometrically and statistically. The simulation develops a 1 MK corona and readily produces loops as seen in Figure 2, showing the underlying magnetogram at the bottom with the calculated intensity in the TRACE 171 A(left)˚ and 195 A(righ˚ t) pass bands. The results have passed several tests, such as the calculated data number counts for the simulated TRACE images agree well with the counts from a typical quiescent active region, and has lately also reproduced the temperature – DEM relation (Peter et al. 2004). In spite of that, the region with low temperatures are possibly not well reproduced, since the optically thick and line cooling in this region is replaced by a Newtonian cooling scheme. Even with this inadequacy, something can probably still be learned, since the cooling most likely is not very different from the cooling on the Sun. Following the field lines beginning in a straight line at X = 14 Mm and Y = 24 – 44 Mm do not show anything that looks like a wine glass (Figure 3) and so if one believes the dynamics of the simulation, then one has to discard the wine-glass picture. 338 Gudiksen

5. Conclusions

With the limited knowledge we have on the magnetic configuration in the layer between the photosphere and the β = 1 surface, there is convincing evidence for a configuration that does not look like the classical picture produced by Giovanelli (1980). The field is much more complicated and this will have an effect on, for instance, previous calculations of wave propagation in this layer. Several open questions then spring to mind from this conclusion. One is whether the deviation from a force free configuration in the photosphere and chromosphere is so small that it can be ignored. If not, then how large a mistake do we make by assuming it to be so, and what length scales are affected. The answers to these questions might well depend on the emerging field configuration as well as the driving velocity field of the photosphere, so maybe the questions cannot be answered in any sensible way. Another question would be what the effect on the chromosphere the magnetic configuration has. Remember that the layer where the magnetic field is not ignorable, for instance β ≤ 10 is most of the upper chromosphere, and therefore the field could change both the dynamics and the emerging radiation, making it crucial to include it when interpreting observations from this part of the atmosphere – a truly daunting task. A final remark is that all is not as hopeless at it seems. Lately instruments have been, and are coming online that can for instance make high resolution, high cadence vector magnetograms from the photosphere and the chromosphere (for instance at the newly upgraded Swedish Solar Telescope), and an instrument that is able to make measure- ments of the coronal magnetic field (at Sac Peak) as well as several space borne observa- tories, such as Solar-B (continuous photospheric vector magnetograms), STEREO (just that – observing structures from two angles), and eventually Solar Orbiter. Theoretical, and observational results from these instruments, should provide interesting answers.

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