Stefan J. Hofmeister1, Dominik Utz1, Stephan G. Heinemann1, Astrid Veronig1,2, Manuela Temmer1
1 Institute of Physics, University of Graz, Graz, Austria
2 Kanzelhöhe Observatory for Solar and Environmental Research, University of Graz, Graz, Austria
It is well known that coronal holes exhibit a magnetic field topology open toward interplanetary space, however, their sources have not been studied in detail. We use HMI’s low-noise line-of-sight magnetograms to analyze the distribution and properties of photospheric magnetic elements below 98 coronal holes, which are the footpoints of both closed loops and open magnetic funnels within coronal holes. The magnetic elements were identified by applying a threshold of ±25 G on the magnetograms and requiring that they contain at least a core of 2×2 pixels above the threshold to remove spurious artefacts originating from noises (Fig. 1).
We find that all magnetic elements follow a power law between their area and magnetic fluxes (cc=0.984), given by
Figure 1| AIA-193 filtergram of a coronal hole observed on July 25, 2013. Green contours outline magnetic elements with dominant coronal hole polarity, blue with non-dominant polarity.
Therefore, their magnetic flux is statistically well determined by their size. Further, by tracking them for ±2days, we find that their lifetimes group them into four categories. The first three categories are given by their half-lives of 14 min, 2.1 h, and 11.7 h, which relate them to the convective motions of granulation, mesogranulation, and supergranulation. By comparing the number of magnetic elements with lifetimes > 4 days in our dataset with the number we expect from supergranulation, we find elements two orders of magnitude more than we should have. Therefore, we define a fourth class of magnetic elements, i.e., long-lived magnetic elements with lifetimes > 40 h.
Figure 2| Number densities of magnetic elements vs. their magnetic field strength. Top left: magnetic elements belonging to the class of granulation. Top right: to mesogranulation. Bottom left: to supergranulation. Bottom right: long-lived magnetic elements.
In Figure 2, for each of the four classes, we show a superposed epoch analysis for their number densities within coronal holes. Thereby, the magnetic field strength of the magnetic elements has been transformed so that positive values represent the dominant coronal-hole polarity. For the short- to medium-lived classes of magnetic elements, i.e., magnetic elements related to granulation, mesogranulation, and supergranulation, there are almost as many magnetic elements with dominant coronal-hole polarity as with non-dominant coronal-hole polarity. Since their polarities are well balanced, their contribution to the open magnetic flux of coronal holes is almost negligible; they are mostly likely the foot points of closed loops. In contrast, the long-lived magnetic elements have almost exclusively the dominant coronal-hole polarity, with an average contribution to the open magnetic flux of 69 %. Thus, they are the main source of open magnetic flux. The remaining percentage of open magnetic flux can be modelled as a weak background magnetic field of 0.2 G to 1.2 G, which is strongly correlated to the magnetic flux arising from the long-lived magnetic elements (cc=0.88). Here, we cannot distinguish whether this background magnetic field is real or an artefact created by the point-spread function of HMI.
Figure 3| Mean magnetic field strength (left) and mean unsigned magnetic field strength (mid) of the overall coronal holes vs. the percentage area the long-lived magnetic elements cover. Percentage unbalanced magnetic versus the mean magnetic field strength of coronal holes (right).
Finally, we compare the properties of the magnetic elements with the mag- netic properties of the overall coronal holes (Fig. 3). We find that the magnetic field strength of coronal holes is fully determined by the percentage area the long-lived magnetic elements cover (cc=0.988):
This relationship follows from the area-flux relationship of individual magnetic elements (Eq. 1), and from them being the main contributor to the open magnetic flux of the overall coronal hole. Further, the mean unsigned magnetic field strength is given by
The first term is set by the noise level in the magnetograms, and the second term describes the increase of unsigned magnetic flux with increasing number of long-lived magnetic elements; the contribution of the short- to medium-lived magnetic elements is negligible. The percentage of unbalanced magnetic flux of coronal holes is defined and given by
It is usually interpreted as the percentage of magnetic flux which is open to the flux which belongs to closed loops. This equation, however, shows that it is only a measure on the magnetic flux arising from the long-lived magnetic elements as compared to the magnetogram noise level, and thus has no reasonable physical meaning.
To conclude, long-lived magnetic elements with lifetimes > 40 h set the magnetic properties of the coronal holes. More details can be found in Ref .
 Hofmeister, S. J., Utz, D., Heinemann, S. G., Veronig, A., & Temmer, M. 2019, A&A, 629, A22
Is there additional evidence for the statements following Figure 2 in this nugget that the shorter-lived bipolar concentrations do not contribute to the open flux?
I ask because it also seems possible to me that even the shorter-lived magnetic elements related to granulation, mesogranulation, and supergranulation may still contribute to the open flux, due to the “flux-stealing” effect presented in Schrijver & Title (2003,ApJ,597,L165). That letter illustrates how mixed-polarity internetwork field concentrations could contribute as much as 50% of the open flux, since in many cases the minority polarity of the internetwork field may connect to the larger-scale unbalanced polarities in the open-flux region (coronal hole), thereby allowing the dominant polarity of these internetwork fields to connect elsewhere, or be open. Additionally, coronal loops often show dynamics on timescales associated with granulation, which also seems to suggest that the associated field lines are not anchored in longer-lived magnetic elements.
I agree that some part of the flux of the large magnetic elements connect to a number of weaker magnetic elements of the non-dominant polarity, i.e., the flux stealing is present. This can be seen by following loops in AIA-171 from the large magnetic elements to the surrounding smaller ones, and looks similar to Figure 1 of Schrijver & Title (2003,ApJ,597,L165). However, almost all foot points of the so-closed loops seem to be in the lanes of the supergranular cells and not in internetwork (at least as we can see them in AIA-171), and comparing the fluxes at the foot points, only a rather small percentage of the flux of the large magnetic elements seems to correspond to closed loops. But of course, we can miss some loops which are not visible in AIA-171.
So, I think at the moment there is no further evidence, and a final answer to where the open field lines are rooted is not apparent. But we are working on it from two perspectives: On the one side, we use artificial magnetograms to check the influence of magnetogram noise on magnetic field extrapolations and on which magnetic field lines are modeled as open, on the other side, we plan to trace the magnetic field lines and fluxes in coronal holes through the chromosphere using high-resolution spectroscopy.