Hermance J. Hagenaar

Sustaining the Quiet Photospheric Network: The Balance of Flux Emergence, Fragmentation, Merging, and Cancellation

The magnetic field in the solar photosphere evolves as flux concentrations fragment in response to sheared flows, merge when they collide with others of equal polarity, or (partially) cancel against concentrations of opposite polarity. Newly emerging flux replaces the canceled flux. We present a quantitative statistical model that is consistent with the histogram of fluxes contained in concentrations of magnetic flux in the quiet network for fluxes exceeding ≈ 2 × 1018 Mx, as well as with estimated collision frequencies and fragmentation rates. This model holds for any region with weak gradients in the magnetic flux density at scales of more than a few supergranules. We discuss the role of this dynamic flux balance (i) in the dispersal of flux in the photosphere, (ii) in sustaining the network-like pattern and mixed-polarity character of the network, (iii) in the formation of unipolar areas covering the polar caps, and (iv) on the potential formation of large numbers of very small concentrations by incomplete cancellation. Based on the model, we estimate that as much flux is cancelled as is present in quiet-network elements with fluxes exceeding ≈ 2 × 1018 Mx in 1.5 to 3 days, which is compatible with earlier observational estimates. This timescale is close to the timescale for flux replacement by emergence in ephemeral regions, so that this appears to be the most important source of flux for the quiet-Sun network; based on the model, we cannot put significant constraints on the amount of flux that is injected on scales that are substantially smaller than that of the ephemeral regions. We establish that ephemeral regions originate in the convection zone and are not merely the result of the reemergence of previously cancelled network flux. We also point out that the quiet, mixed-polarity network is generated locally and that only any relatively small polarity excess is the result of flux dispersal from active regions.

Ephemeral Regions on a Sequence of Full-Disk Michelson Doppler Imager Magnetograms

Ephemeral regions are small-scale, bipolar regions of magnetic field, emerging all over the solar surface. As structures ephemeral regions are short-lived; they can be recognized for 4.4 hr on average. This paper examines a 3.5 day sequence of full-disk Michelson Doppler Imager (MDI) magnetograms in order to estimate the importance of ephemeral regions to the total magnetic flux budget on the sun. The data were taken in 1997 October, which was around the minimum of solar cycle 22, and early in cycle 23. An algorithm was developed to automatically recognize ephemeral regions on this sequence of magnetograms. Assuming uniform emergence over the entire solar surface, the total amount of flux emerging in ephemeral regions is estimated to be 5 × 1023 Mx-1. This rate of emergence is sufficient to replace the magnetic field in quiet sun in 14 hr. In total, 38,000 ephemeral regions are found, with an absolute flux Φ in the range (2.6-407) × 1018 Mx. The distribution function of their fluxes follows an exponential with an average of 11.3 × 1018 Mx. This relatively low flux content may be due to the fact that these ephemeral regions are detected before they have reached a maximum. After their first recognition, they increase in flux with a rate of typically dΦ/dt = 1.6 × 1015 Mx s-1. Only 60% of the ephemeral regions are found to have the orientation expected in cycle 22. After emergence, the outer borders of the ephemeral regions expand from a size of 8.9 Mm, with a velocity of 2.3 km s-1. No particular pattern can be recognized from a map of all locations of flux emergence. From a χ2 test it is found that the emergences occur randomly, on a scale below 20 Mm. On larger scales some order is found, but its origin remains unknown.

Dispersal of Magnetic Flux in the Quiet Solar Photosphere

We study the random walk of magnetic flux concentrations on two sequences of high-resolution magnetograms, observed with the Michelson Doppler Imager on board SOHO. The flux contained in the concentrations ranges from |Φ|=1018 Mx to |Φ|=1019 Mx, with an average of |Φ|=2.5×1018 Mx. Larger concentrations tend to move slower and live longer than smaller ones. On short timescales, the observed mean-square displacements are consistent with a random walk, characterized by a diffusion coefficient D(t 30 ks)=200-250 km2 s−1, approaching the measurements for a five-day set of Big Bear magnetograms, D250 km2 s−1. The transition between the low and large diffusion coefficients is explained with a model and simulations of the motions of test particles, subject to random displacements on both the granular and supergranular scales, simultaneously. In this model, the supergranular flow acts as a negligible drift on short timescale, but dominates the granular diffusion on longer timescales. We also investigate the possibility that concentrations are temporarily confined, as if they were caught in supergranular vertices, that form short-lived, relatively stable environments. The best agreement of model and data is found for step lengths of 0.5 and 8.5 Mm, associated evolution times of 14 minutes and 24 hr, and a confinement time of no more than a few hours. On our longest timescale, DSim(t>105)→285 km2 s−1, which is the sum of the small- and large-scale diffusion coefficients. Models of random walk diffusion on the solar surface require a larger value: DWang=600±200 km2 s−1. One possible explanation for the difference is a bias in our measurements to the longest lived, and therefore slower concentrations in our data sets. Another possibility is the presence of an additional, much larger diffusive scale.

The Distribution of Cell Sizes of the Solar Chromospheric Network

This paper studies the cellular pattern of the supergranular network. We present an algorithm to draw a surface-filling cell pattern on an uninterrupted two-day sequence of Ca II K filtergrams with a 1 nm bandpass. The 60° × 40° field of view contains both quiet and enhanced network and plages. The algorithm uses a threshold-independent method of steepest descent on spatially smoothed and time-averaged images. We determine the distribution function of cell areas and find a broad, asymmetric spectrum of areas. The distribution is found to be invariant for different spatial smoothings if the cell areas are normalized to a unit mean. It is this invariance that leads us to believe we have determined the intrinsic distribution of cell areas. Extrapolation of the average cell size to zero spatial smoothing yields a characteristic cell diameter of L = 13-18 Mm. This is roughly half the generally quoted supergranular length scale L ≈ 32 Mm as determined with autocorrelation methods. The difference in characteristic cell size reflects the application of a different measurement method: the autocorrelation method as used by Simon & Leighton and others is preferentially weighted towards relatively large cells. We find no significant dependence of cell size on local magnetic flux density.

On the Patterns of the Solar Granulation and Supergranulation

We study the cellular patterns of the white light granulation and of the chromospheric Ca II K supergranular network. We apply a gradient-based tessellation algorithm to define the cell outlines. The geometry of the patterns formed by the associated granular and supergranular flows are very similar, in spite of the substantial difference in length scale. We compare these patterns to generalized Voronoi foams and conclude that both convective patterns are very nearly compatible with an essentially random distribution of upflow centers, with the downflow boundaries determined by the competing strengths of outflows of neighboring upwellings. There appears to be a slight clustering in upflow positions for the granulation, consistent with the granular evolution. This slight preference for large granules to be surrounded by somewhat smaller ones makes the granular and supergranular patterns differ enough to allow a correct identification in three out of four cases by eye. The model analogy suggests that the range in outflow strengths is remarkably small. The patterns appear to be rather insensitive to the details of the competing forces that establish the pattern of the downflow network: similar patterns result under very different conditions, so that little can be learned about the details of the forces involved by studying the geometry of these patterns only.

Modeling the Distribution of Magnetic Fluxes in Field Concentrations in a Solar Active Region

Much of the magnetic field in solar and stellar photospheres is arranged into clusters of ‘flux tubes’, i.e., clustered into compact areas in which the intrinsic field strength is approximately a kilogauss. The flux concentrations are constantly evolving as they merge with or annihilate against other concentrations, or fragment into smaller concentrations. These processes result in the formation of concentrations containing widely different fluxes. Schrijver et al. (1997, Paper I) developed a statistical model for this distribution of fluxes, and tested it on data for the quiet Sun. In this paper we apply that model to a magnetic plage with an average absolute flux density that is 25 times higher than that of the quiet network studied in Paper I. The model result matches the observed distribution for the plage region quite accurately. The model parameter that determines the functional form of the distribution is the ratio of the fragmentation and collision parameters. We conclude that this ratio is the same in the magnetic plage and in quiet network. We discuss the implications of this for (near-)surface convection, and the applicability of the model to stars other than the Sun and as input to the study of coronal heating.

ERRATUM: “THE DEPENDENCE OF EPHEMERAL REGION EMERGENCE ON LOCAL FLUX IMBALANCE” (2008, ApJ, 678, 541)

We have discovered an error in the labeling of Figure 5. The importance of the figure is to indicate the dependence of flux emergence on local flux (im-) balance. However, the scales of the figures were incorrect, causing a discrepancy between Table 2 and Figure 5(a). The corrected Figure 5 appears below. The change does not affect the conclusion.