Validation & Testing of the CROBAR 3D Coronal Reconstruction Method with a MURaM simulation

I use the same coronal MURaM simulation also reported in Cheung et al. (2019); Malanushenko et al. (2022), however I use a somewhat earlier time in the simulation (The time of the snapshot may be referenced by its unique index of 100000) during which the volume is less active. The CROBAR approach is less likely to be suited to times of strong plasma dynamics when the field-alignment of the plasma is less certain and the field is less likely to be force free. This time interval in the simulation still contains the complex structures pointed out by Malanushenko et al. (2022) and shows the sort of complexity associated with solar active regions. It is important to note that there some differences with a typical solar active region, however:

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  The region is significantly smaller than the long-lasting active regions most often the focus of solar physics research, at roughly $100$ by $50$ by $50$ Megameters, or fewer than 200 pixels across as viewed by SDO/AIA. Many active regions are several times this size in linear extent.

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  The boundary conditions are periodic, so that the region behaves as if it has an identical clone of itself on each side rather than being relatively isolated.

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  The simulation performs a number of approximations to deal with aspects that are to computationally demanding to incorporate (such as extremely high Alfven wave speeds in some regions), and assumes the chromosphere is LTE.

For more discussion of the details of the simulation, see Rempel (2017); Cheung et al. (2019); Malanushenko et al. (2022). Figures showing the appearance of the simulation will be shown in connection with the reconstructions, as the narrative develops.

As previously mentioned, this paper treats only the plasma emission part of the problem. The emphasis is on the question of whether field aligned emission can reproduce coronal emission, at all, regardless of the quality of the magnetic field skeleton. There is little point in covering the refinement of the magnetic field skeleton before this question has been addressed, as Malanushenko et al. (2022) has illustrated. Therefore, I use as the skeleton the magnetic field contained in the MURaM simulation rather than (for example) attempting to use and refine a linear force-free field. I intend to address that topic in a future paper, and have already carried out some work in that direction with promising results.

I traced 10,500 field lines through the volume, using the MURaM field vectors. The tracing was done with the 2/3 order Runge-Kutta scheme included in scipy’s integrate.solve_ivp method. 10,000 of those initial points were at the photosphere level and their location was randomly chosen with a weight based on the vertical field strength at the photospheric level. The other 500 initial points were uniformly distributed through the volume and then traced in both directions until they reached the photosphere or the edges of the cube. I removed field lines from the set which were too short in either height (less than one voxel, 0.064 Mm) or overall length (less than 8 voxels, 0.512 Mm). This is referenced to the voxel grid used for associating points in the volume with field lines, which was chosen to have 0.064 Mm grid size in each direction rather than the $192\times 192\times 64$ Mm grid resolution used in the MURaM simulation. This choice was made to ensure full resolution of the simulation and also because the computational load if CROBAR is light. The voxel grid used by CROBAR therefore has just under 1 billion voxels, requiring 3 GB of RAM (2 GB for the array indicating the loop ID of each voxel, and 1 GB for the array indexing its length along the loop).

Images of the seed points for tracing fieldlines and the vertical field component at the nominal photospheric level (8 Mm, 125 voxel heights, compare Rempel, 2017) are shown in Figure 1. The boundary conditions are at least nominally circular, and I wanted to maximize the number of complete field lines (footpoint to footpoint), so I shifted the cube 128 voxels to the left compared to its usual presentation, placing the negative polarity at cube center. This makes nearly all loops in the cube complete, since most large-scale connections in the cube are between the positive polarity/ies and the negative polarity/ies. I also padded the cube on the top and bottom (using the circular boundary conditions) by 32 pixels, in a further effort to avoid edge effects. The field-aligned regions used for the reconstruction are shown in Figure 2, using both a projection through with the voxels of 1 in 20 of the regions set to one, as well as a set of orthogonal slices through salient locations in the cube.

The padded voxels mentioned above are omitted from the reconstruction, since they contain incomplete field lines. Similarly, I noticed a small edge discontinuity in the cube at 128 voxels, so I only used the right-hand-side (rightmost 256 voxels) of the cube for the reconstructions; emission from voxels outside the resulting $49.152\times 49.152\times 41.152$ Mm subsection of the MURaM cube is set to zero in both the forward modeling and the CROBAR reconstructions. Emission from outside this subsection will not be shown in subsequent figures.

One part of the forward and inverse modeling that is essential to consider is the transition from optically thin emission to optically thick in whatever part of the solar spectrum is being observed. Although coronal emission is generally described as optically thin, at some point it transitions to being optically thick; otherwise we could see through the solar disk in these passbands!

This is complicated by the simplifying assumptions made in the MURaM simulations. While necessary for computational tractability, they result in high-density cusps at the base of the observing region which can appear to dominate the emission, unlike real solar passband images. CROBAR does not accurately reproduce these, which leads to issues with the reconstructions. To avoid these issues, I set a minimum height cutoff of 2.5 Megameters above what I take to be the photospheric height, which in turn is 8 Megameters (125 voxel heights) above the base of the simulation. Above this height, I approximate optical depth attenuation with a local extinction factor $e^{-\tau(z)}$ (compare Mok et al., 2016) where $\tau(z)$ which is proportional to the vertical integral of the density. Naturally, this is inexact for non-vertical lines of sight. The constant of proportionality is set so that optical depth unity occurs at a column density of $3\times 10^{19}\mathrm{cm}^{-2}$.

These issues do also suggest that CROBAR is not as readily suited to passbands or lines which have a significant contribution from non-optically thin emission at the footpoints, although I expect such difficulties can be mitigated with further refinement and iteration. This is not simply a question of where the temperature response function is large, however; because emission is proportional to temperature response times density squared ($R(T)n^{2}$), one must also consider the density where optically thick emission becomes significant. The condition results in the following inequality:

if the emission from optically thin temperatures $T(\tau<<1)$ is to dominate over the optically thick emission $T(\tau=1)$. If the ideal gas law holds and the pressure scale height is large, this becomes

In other words, the $R(T)$ at optically thin temperatures must be greater than at $\tau=1$ temperature by a factor of the square temperature ratio $T(\tau<<1)/T(\tau=1)$. This may have underappreciated implications for the temperatures of other features observed in the corona such as the moss; The temperature of the most visually apparent features at low heights may be much cooler than the shape of the temperature response function alone would indicate.

The temperature response function for this exercise was the power with law index $2$ mentioned in Plowman (2021). This is the ideal response function for this purpose because the emission profile of each region is a function only of pressure and dependence on specific details of the region’s temperature profiles is minimized. The linearity assumption is therefore most nearly satisfied. As Plowman (2021) noted, this kind of response function can be synthesized from a DEM, and it is also very similar to some lower X-Ray response functions such as from the Hinode X-Ray Telescope (XRT Golub et al., 2007) or Yohkoh Soft X-Ray Telescope (SXT Tsuneta et al., 1991). I have also noticed that they tend look quite similar to images from AIA 335 Å, suggesting that a large fraction of coronal emission comes from temperatures where that passband has a scaled powerlaw index of 2. This is a potentially interesting result, which I intend to investigate further. For the reconstructions in this paper, I will consider viewpoints of $0^{\circ}$ (overhead) alone, $0^{\circ}+60^{\circ}$, and $0^{\circ}+90^{\circ}$ ($90^{\circ}$ corresponds to a typical limb view of the region). those are each shown in Figure 3 along with an equivalent overhead view with the AIA 335 Å temperature response function (The latter shows more ‘cuspy’ low-lying emission than the $T^{2}$ channel, more so than is typical with real AIA 335 Å data, so I expect those features are due at least in part to the simplifying assumptions of MURaM previously mentioned). I also show a longitudinal slice through the region, illustrating that this emission contains the interesting structures noted by Malanushenko et al. (2022). Each of these views of the MURaM cube will also be shown alongside the reconstructions.

Although CROBAR can provide 3D information from just a single perspective, the results shown here point out the importance of additional viewpoints and quantify the improvement from adding another viewpoint. They also indicate that both 60 and 90 degrees as additional perspectives provide similar improvements. This suggests that a ’drifter’ spacecraft, moving from $\sim 45$ to 90 degrees over its lifetime, could provide a similar degree of reconstruction quality to one which was ’parked’ (e.g., at a Lagrange point) over its entire lifetime. This would be similar to the orbits of the STEREO spacecraft. Orbits like that of Solar Orbiter, while usable for science work, would be less desirable from a space weather standpoint due to inconsistent temporal coverage.

Most coronal space observations have focused on EUV lines or passbands with relatively low temperatures and narrow temperature coverage. However, some of the considerations discussed in Section 2.2 indicate that these are not the best suited to the coronal 3D reconstruction problem. Instead, a passband with a temperature response that is a power law (or at least monotonic), peaked at high temperatures (hotter than the plasma being observed) is ideal. This suggests an X-ray imager (a la Yohkoh SXT or Hinode XRT) or perhaps a relatively broad-band EUV imager if a suitable wavelength window can be found. The other option would be to use DEMs, although that requires more lines/passbands (more complexity, weight, and telemetry) and must still include lines or passbands with high temperature response. That said, I have seen surprisingly promising results applying CROBAR to AIA passbands such as 335 Å and 171 Å, or the STEREO passbands (STEREO does not have much in the way of high temperature coverage, see comment below). I am still investigating this and it will be the topic of a future publication. Similarly for data from Solar Orbiter García Marirrodriga et al. (2021), particularly the SPICE spectrograph (Spice Consortium et al., 2020).

The short version of the above two paragraphs is that an ‘optimal’ cost/benefit deep space mission to take advantage of CROBAR would include either an imager with soft-X-Ray-like temperature response functions (2 distinct channels) or an AIA like instrument with at least 5 channels and high temperature coverage at least as good as what’s provided by the 335 Å and 94 Å channels (these are essential for AIA to cover the critical temperature range from $10^{6.4}to10^{6.8}$ Kelvin, where many active region field lines have their temperature peaks). It should also observe from between 30 and 90 degrees. No spacecraft has done this to date; STEREO is the closest but its high temperature coverage is not well suited to CROBAR and the lifespan of the remaining spacecraft is highly uncertain.

The utility of CROBAR as a starting point, giving a three-dimensional emission structure corresponding to a coronal volume from just one or two vantage points, is substantial. It provides a viable path forward for future improvement: with it we can refine and optimize our models of the field with guidance from coronal emission observations, we can add refined models for the field aligned emission in accordance with modelers, and experiment with adding more detailed physics to the framework as opportunity and insight present themselves. Whereas without it we had been limited to speculating about how our 2D images connect to the actual 3D volume.

This paper has focused on testing for the field-aligned reconstruction concept against the MURaM simulations, but I have already found significant success with initial implementation of the refinements just mentioned in the context of real solar data. I have already implemented a basic tunable field model, in the form of a nonlinear force-free-field, and applied it to CROBAR reconstructions with AIA and STEREO data with very promising results. These will be covered in an upcoming publication.