Hinode spectropolarimetry revealed that the quiet solar internetwork is blanketed by transient, nearly horizontal magnetic fields whose apparent transverse flux density at disk center exceeds the longitudinal one by a factor of about five, plausibly constituting the 'hidden turbulent flux' inferred from Hanle-effect depolarization. Lites (2009) synthesized the evidence that these horizontal internetwork fields are distributed uniformly over the Sun and hence are likely generated by a local dynamo on granular scales, and listed three unresolved issues: the unexplained center-to-limb behavior of the Fe I 630 nm polarization signals, the unknown solar-cycle variation of the field population, and the unelucidated generation and sustenance mechanism. This problem asks for the origin and sustenance of these fields at a quantitative level: not merely whether a local dynamo - rather than recycling of decaying active-region flux or emergence from the global dynamo - is responsible, but a validated quantitative mechanism by which small-scale dynamo action driven by granular convection generates and sustains them at solar conditions (low magnetic Prandtl number, realistic stratification). Later work has since addressed the first two of the issues listed by Lites (2009); it is this third issue that remains open and that this problem poses.

Background

The quiet Sun - the solar surface outside sunspots and active regions - is covered by magnetic fields organized on small scales. Its 'network' consists of concentrated, vertically oriented kilogauss flux tubes outlining supergranular cells (large-scale convective flow cells), while the 'internetwork' (IN) inside the cells hosts weaker, mixed-polarity, rapidly evolving fields. Magnetic fields are measured through the polarization they imprint on spectral lines: the Zeeman effect (the splitting and polarization of a spectral line in a magnetic field) makes the circular polarization signal (Stokes VV) sensitive to the line-of-sight field component and the linear polarization signals (Stokes QQ and UU) sensitive to the transverse component; the apparent longitudinal and transverse flux densities are the field strengths inferred from these signals under simple assumptions. The Hanle effect (the modification of scattering polarization by weak magnetic fields) is sensitive to tangled, unresolved fields of roughly 1-100 G - the 'hidden turbulent flux' of the quiet Sun, estimated from Hanle depolarization of lines such as Sr I 4607 Å to be of order 100 G on average by Trujillo Bueno, Shchukina and Asensio Ramos (2004).

The Hinode satellite's Solar Optical Telescope Spectro-Polarimeter (SOT/SP), providing seeing-free 0.3-arcsec spectropolarimetry in the Fe I 630.1/630.2 nm line pair of neutral iron, revealed that the internetwork is blanketed by transient, nearly horizontal fields - the horizontal internetwork fields (HIFs): the apparent transverse flux density at disk center exceeds the longitudinal one by a factor of about five (Lites et al. 2008). Ground-based full-disk magnetograms independently show a 'seething' horizontal field whose fluctuations increase toward the limb (Harvey et al. 2007), and Lites (2009) argued that the two are the same phenomenon, distributed uniformly over the solar surface including the poles, and suppressed only in strong plage (enhanced magnetic regions outside sunspots; Ishikawa et al. 2008). Because a population tied to the global dynamo that drives the solar activity cycle should cluster at active latitudes and vary with the cycle, this uniformity pointed to a local dynamo operating on the scale of granulation (the roughly 1000-km convective cells of the solar surface) - a 'small-scale dynamo' (SSD), in which turbulent convective motions amplify magnetic field through stretching, folding, and twisting of field lines, and which can operate when the magnetic Reynolds number RmR_{\mathrm{m}} (the ratio of advective to diffusive magnetic transport) exceeds a critical value. A central physical concern is the magnetic Prandtl number PmP_{\mathrm{m}} (the ratio of viscosity to magnetic diffusivity): simulations necessarily run near Pm≈1P_{\mathrm{m}}\approx 1, whereas in the solar photosphere Pm∼10−5P_{\mathrm{m}}\sim 10^{-5}, and whether the dynamo survives at such low PmP_{\mathrm{m}} in a stratified, radiatively cooled plasma with open boundaries is itself part of the question.

Lites (2009) closed by listing three unresolved issues, of which this problem takes up the one that remains open. First, the center-to-limb (C-L) behavior of the linear polarization signals in the Fe I 630 nm lines was not explained by the simple models presented in Lites et al. (2008): both the apparent longitudinal and transverse flux densities decrease toward the limb, contrary to the naive expectation that the inclined horizontal fields should boost the longitudinal signal. Second, nothing was known about possible solar-cycle variations of the HIF population, since the data were taken near solar minimum. Third, although the evidence supported a local dynamo origin, the mechanisms generating and sustaining these turbulent fields on granular scales remained to be elucidated by theoretical models and numerical simulations.

Problem Statement

Supply a validated quantitative mechanism for the generation and sustenance of the ubiquitous horizontal internetwork (turbulent) magnetic fields of the quiet solar photosphere - in particular, how small-scale dynamo action driven by granular convection generates and sustains them at solar conditions (low magnetic Prandtl number, realistic stratification) - or a demonstration that no such mechanism exists.

Of the three issues listed by Lites (2009), the center-to-limb behavior of the Fe I 630 nm Zeeman polarization signals and the solar-cycle variation of the field population have since been addressed and are treated here as background rather than as the deliverable; the issue this problem poses is the third. Validation is required against the observed properties of the horizontal internetwork fields - the disk-center ratio of apparent transverse to longitudinal flux density and the spatial distribution and evolutionary timescale of the field population - to within the uncertainties of the published spectropolarimetric measurements; a qualitative sketch of a plausible mechanism does not count as validation. Reproducing the center-to-limb variation of the Fe I 630 nm signals is not required, that issue being treated as background above.

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