Hinode/SOT observations of the Sun's polar region revealed a photospheric magnetic landscape dominated by vertically oriented kilogauss flux tubes whose cross-sectional area expands by up to a factor of about 345 between the photosphere and the lower corona. Tsuneta et al. (2008) conjectured that these rapidly expanding tubes act as efficient 'chimneys' for Alfvén waves — transverse magnetohydrodynamic oscillations that propagate along magnetic field lines — because the strong areal expansion softens the vertical Alfvén-speed gradient, lowers the wave-reflection cutoff frequency, and thereby lets more photospherically generated wave energy reach the corona and accelerate the fast solar wind. The conjecture has never been quantitatively tested against the observed polar geometry: existing estimates of how much Alfvén-wave flux crosses the low solar atmosphere span roughly 1% (quiet-Sun expanding-tube calculations) to about 85% (wave-driven wind simulations with generic open-field geometry), leaving the central quantity unpinned. The problem is to determine, by forward modeling and/or observation, the transmission and reflection of Alfvén waves through the measured expanding geometry of the polar kilogauss flux tubes and to decide whether the transmitted energy flux meets the energy budget of the fast solar wind emanating from polar coronal holes.

Background

The fast solar wind — the steady, high-speed (~750 km/s) plasma outflow emanating from coronal holes, the dark, low-density, unipolar-magnetic regions of the solar corona such as those covering the solar poles near activity minimum — must be powered and accelerated by some combination of wave pressure and reconnection-driven jets. Alfvén waves, transverse incompressible oscillations propagating along magnetic field lines at the Alfvén speed, are a leading candidate: they are ubiquitously generated by photospheric convection and carry more than enough energy if a sufficient fraction survives the journey through the steeply stratified low atmosphere. The obstacle is reflection: where the Alfvén speed varies rapidly with height, upward-propagating waves with frequencies below a cutoff set by the Alfvén-speed gradient are largely reflected back down, so only a poorly known fraction of the photospheric wave flux reaches coronal heights.

The relevant magnetic skeleton at the poles was measured by the Hinode Solar Optical Telescope (SOT). Tsuneta et al. (2008, ApJ 688, 1374) found that the polar photosphere outside ephemeral regions — small, short-lived magnetic bipoles — is carpeted with patchy, vertically oriented flux tubes with intrinsic (filling-factor-corrected) field strengths of order kilogauss; the filling factor — the fraction of a resolution element actually occupied by magnetic field, as opposed to the spatially averaged value — peaks near f∼0.15f\sim0.15, and only 10.5% of pixels met the signal-to-noise criterion of their Milne–Eddington inversions — a standard technique that fits a simplified model atmosphere to the measured polarized spectral-line profiles to recover the magnetic field vector — so the reported fluxes are lower limits. From the observed fanning of the field vectors and a total-area comparison between the polar field of view and the combined cross-section of the vertical flux tubes, they estimated an areal expansion of up to roughly a factor 345 for these tubes between the photosphere and the lower corona: the tubes fan out rapidly and merge into the open field of the polar coronal hole. From this they articulated the chimney conjecture. Rapid areal expansion near the photosphere–corona boundary means the magnetic field strength, and hence the Alfvén speed, varies more gently with height than in a slowly expanding tube; the Alfvénic scale height lengthens, the cutoff frequency for reflection drops, and photospherically excited waves pass into the corona more efficiently. The tube is thus a 'chimney' for Alfvén waves, and the polar holes are efficiently plumbed for wave-driven wind acceleration.

The theoretical machinery for testing this exists in nearby contexts. Reflection and transmission of torsional Alfvén waves in expanding flux tubes was computed by Soler et al. (2017, ApJ 840, 20), who showed that larger expansion factors shift the transmission window to lower frequencies, exactly the ingredient the chimney argument invokes; applying related modeling to a quiet-Sun tube with a 1 kG photospheric field, Soler et al. (2019, ApJ 871, 3) found that only about 1% of an injected 10710^7 erg cm−2^{-2} s−1^{-1} torsional-wave flux reaches coronal heights. At the other extreme, one- to three-dimensional magnetohydrodynamic simulations of wave-driven wind (Suzuki & Inutsuka 2006; Matsumoto & Suzuki 2012, 2014; Matsumoto 2020), in which footpoint-driven transverse fluctuations reflect, cascade, and heat the open corona, reproduce a transonic fast wind and implicitly allow most wave flux into the corona; Morton et al. (2023, Reviews of Modern Plasma Physics) review this spread — from roughly 95% reflection at the transition region, the sharp interface between the chromosphere and the corona, in some treatments, to about 85% transmission in wind simulations, to about 1% in recent three-dimensional incompressible calculations — and note that the radial fine structure of polar coronal holes remains little-studied. Observationally, propagating Alfvénic motions are detected in polar coronal holes (Banerjee et al. 2009) and carry sufficient energy for the fast wind in the corona (McIntosh et al. 2011, Nature 475, 477), but no observation ties the coronal wave flux to the photospheric flux tubes that launched it or measures the transmission fraction. What is missing is the conjunction: a quantitative determination of Alfvén-wave transmission through the actually measured polar flux-tube geometry — kilogauss intrinsic fields, areal expansion up to ~345, coronal-hole stratification — and a comparison of the transmitted flux against the fast-wind energy budget.

Problem Statement

Quantitatively test the chimney conjecture of Tsuneta et al. (2008): determine, by forward magnetohydrodynamic modeling and/or observational measurement, the reflection and transmission of Alfvén waves propagating through the measured expanding geometry of the Sun's polar photospheric flux tubes — intrinsic kilogauss fields with filling factor f∼0.15f\sim0.15, areal expansion of up to ≈345\approx345 between the photosphere and the lower corona, and polar coronal-hole stratification — and decide whether the wave energy flux transmitted through these tubes into the corona is sufficient, in magnitude and in spectral content, to power and accelerate the fast solar wind. A complete answer must (i) compute or measure the height- and frequency-resolved transmission coefficient for the observed expansion profile — the height-resolved cross-sectional expansion of the polar kilogauss tubes anchored to the measured quantities (intrinsic kilogauss fields, filling factor f∼0.15f\sim0.15, and integrated areal expansion of up to ≈345\approx345 between the photosphere and the lower corona; Tsuneta et al. 2008 report only these aggregate quantities, so the height-resolved reconstruction and its systematic error must be constructed and stated as part of (i), and a polar-specific expanding-tube profile anchored to these measured values satisfies this requirement whether it is obtained by re-inverting Hinode/SOT polar magnetograms or by a parametrized model whose field strength follows the measured area expansion) — rather than for a generic open tube, i.e. a uniform or solar-typical super-radial geometry that is not anchored to the polar measurements, (ii) state the photospheric driver amplitude and spectrum assumed or measured, with any assumed driver required to be consistent with photospheric wave-amplitude measurements in the polar region rather than freely tuned and the sensitivity of the transmitted flux and of the sufficiency verdict to this input quantified, and (iii) compare the transmitted flux and wave-pressure gradient against the mass flux and energy budget of the fast wind from polar coronal holes, thereby confirming or refuting that the rapid expansion acts as an efficient wave chimney.

References

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