The internetwork (IN) — the weak, short-lived magnetic patches inside supergranular cells — is the largest known source of fresh magnetic flux reaching the solar photosphere, yet published estimates of its flux emergence rate (FER) disagree by roughly two orders of magnitude: from about 3.7×10243.7\times10^{24} Mx day−1^{-1} (120 Mx cm−2^{-2} day−1^{-1}; Gošić et al. 2016) to 3.8×10263.8\times10^{26} Mx day−1^{-1} (Zhou et al. 2013) summed over the whole Sun (the maxwell, Mx, is the cgs unit of magnetic flux). The spread traces to instrumental detection limits (features below roughly 6×10156\times10^{15} Mx are invisible to the Hinode solar telescope, while the Imaging Magnetograph eXperiment (IMaX) on the Sunrise balloon-borne telescope reaches 9×10149\times10^{14} Mx), cadence, differing operational definitions of 'emergence', tracking rules, and spectral-line differences, and it is compounded by the possibility that some apparent appearance is only coalescence of pre-existing undetected flux. The problem is to determine the internetwork FER in a technique-independent way — through cross-dataset reanalysis under a common emergence definition and uniform detection thresholds, and/or ground-truth validation of the competing methods on magnetohydrodynamic (MHD) simulations containing a working small-scale dynamo — so that the spread among published estimates is quantitatively explained and a converged rate with a systematic uncertainty budget is delivered. The result directly constrains the efficiency of a near-surface small-scale dynamo and the balance of the photospheric magnetic flux budget.

Background

Most of the solar surface is free of sunspots and plage; this 'quiet Sun' carries magnetic structure at three scales: the network (kilogauss concentrations outlining supergranular convection cells), ephemeral regions (small bipolar regions of flux near 101910^{19} Mx living a few hours), and the internetwork (IN) — weak, dynamic patches inside supergranular cells that typically live less than 10 minutes. IN fields are studied in magnetograms, i.e. maps of the photospheric line-of-sight magnetic field, where each identified feature carries an unsigned magnetic flux measured in maxwells (Mx; 1 Mx =10−8=10^{-8} Wb). Because IN features appear and disappear continuously, the IN dominates the fresh flux brought to the surface; the relevant quantity is the flux emergence rate (FER), the total unsigned flux appearing per unit area per day (Mx cm−2^{-2} day−1^{-1}) or summed over the whole Sun (Mx day−1^{-1}).

The FER is not directly observable; it must be inferred under an operational definition of 'emergence'. Common approaches are: (i) tracking every detected feature from appearance to death and summing its flux, separating unipolar appearances, bipolar appearances (the two footpoints of an emerging magnetic loop), and flux gained after birth by features born through splitting or merging; (ii) counting appearing bipolar pairs; and (iii) fitting a power law to the frequency distribution of emergence events over feature flux and extrapolating below the detection limit. Each approach depends on a detection threshold (features below a total flux limit are invisible), a temporal cadence, and tracking rules, and each instrument adds its own spatial resolution, polarimetric sensitivity, and spectral line (different lines form at different atmospheric heights and respond differently to inclined fields).

Published estimates disagree by about two orders of magnitude. Zirin (1987) estimated about 102410^{24} Mx day−1^{-1}. Thornton & Parnell (2011), from Hinode/SOT (Solar Optical Telescope) data, fit a single power law of index ≃−2.7\simeq-2.7 over 101610^{16}–102210^{22} Mx and found 32–470 Mx cm−2^{-2} day−1^{-1} depending on which of three emergence-detection methods (bipole comparison, tracked bipoles, tracked clusters) was used, adopting 450 Mx cm−2^{-2} day−1^{-1} (3×10253\times10^{25} Mx day−1^{-1}) as their global value. Zhou et al. (2013), combining the total IN flux density measured by Hinode with a 3-minute mean feature lifetime, obtained 3.8×10263.8\times10^{26} Mx day−1^{-1}. Gošić et al. (2016), tracking individual features through 38 h of deep Hinode/NFI (Narrowband Filter Imager) magnetograms, measured 120±3120\pm3 Mx cm−2^{-2} day−1^{-1} (3.7×10243.7\times10^{24} Mx day−1^{-1}). Smitha et al. (2017), using Sunrise/IMaX balloon data that resolve features down to 9×10149\times10^{14} Mx — nearly an order of magnitude fainter than the Hinode limit of roughly 66–6.5×10156.5\times10^{15} Mx — measured 1100 Mx cm−2^{-2} day−1^{-1} (6.6×10256.6\times10^{25} Mx day−1^{-1}), with features of flux ≤1016\leq10^{16} Mx contributing most of the flux, and showed that the exact technique can change the result by up to two orders of magnitude even on similar data.

Two deeper issues compound the observational scatter. First, apparent appearance is not necessarily genuine emergence: Lamb et al. (2008) showed that quiet-Sun flux often appears to originate unipolarly, which is incompatible with ∇⋅B=0\nabla\cdot\mathbf{B}=0, and proposed that such flux becomes visible through the coalescence of pre-existing flux too weak and dispersed to detect; apparent appearance rates would then partly recycle old flux rather than inject new flux, biasing every tracking-based FER upward. Second, the physical stake of the FER is its role as a probe of the near-surface small-scale dynamo — a turbulent dynamo acting in and just below the photosphere, distinct from the global dynamo that produces active regions: the amount of genuinely new flux appearing at the smallest scales measures how efficiently such a dynamo operates, and the photospheric flux budget requires flux removal (cancellation, submergence, in-situ fading) to balance emergence at a comparable rate.

Problem Statement

Determine the flux emergence rate (FER) of the quiet-Sun internetwork in a way that is independent of the measurement technique, and quantitatively account for the spread of published values. A complete answer must achieve one coherent objective comprising the following linked requirements:

  1. Fix an operational definition of 'flux emergence' that distinguishes genuinely new magnetic flux crossing the photosphere from the coalescence of pre-existing unresolved flux, stating explicitly how unipolar appearances, bipolar appearances (loop footpoints), and post-birth flux growth of features born through splitting or merging are counted, without double counting.
  2. Apply this definition with uniform detection thresholds and comparable tracking rules across existing datasets (e.g., Hinode/NFI, the Hinode Spectropolarimeter (SP), and Sunrise/IMaX) and/or validate the competing published measurement methods against ground truth in three-dimensional MHD simulations containing a working small-scale dynamo, by degrading synthetic magnetograms to each instrument's actual spectral line, resolution, sensitivity, and cadence.
  3. Quantitatively attribute the discrepancy among the published estimates — from 3.7×10243.7\times10^{24} Mx day−1^{-1} (120 Mx cm−2^{-2} day−1^{-1}; Gošić et al. 2016) through 3×10253\times10^{25} and 6.6×10256.6\times10^{25} Mx day−1^{-1} (Thornton & Parnell 2011; Smitha et al. 2017) to 3.8×10263.8\times10^{26} Mx day−1^{-1} (Zhou et al. 2013) — to named, measured systematic effects, and either reconcile the estimates within quantified uncertainties or demonstrate that specific published values are biased by identified, quantified errors.
  4. Deliver a best estimate of the internetwork FER (per unit area and whole-Sun) with a systematic uncertainty budget, stating the residual range compatible with the evidence and its implication for the efficiency of a near-surface small-scale dynamo and for the balance between flux emergence and flux removal in the photospheric flux budget.

The problem is solved when the roughly two-orders-of-magnitude spread among existing estimates is explained quantitatively, and the competing techniques applied under the common protocol agree within their stated uncertainties.

References

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