Energy goes up... but doesn't come back down! Photospheric energy drives atmospheric heating

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Introduction

As noted in [a recent Nugget], it was recognized more than 70 years ago (Ref. [1]) that the temperature of the solar corona is over a million kelvin, much hotter than underlying photosphere, which is near 6000 kelvin. This is the crux of the coronal heating problem: How is the outer atmosphere of the Sun heated to much higher temperatures than its photosphere?

The second law of thermodynamics implies that the energy required to maintain coronal temperatures cannot flow outward as heat. (Hot material might flow upward to heat the corona from below --- but this energy transport must be non-thermal, e.g., spatially coherent jets of heated plasma like those reported by Ref. [2]). The situation is similar for the 10⁴ K chromosphere, which is also substantially hotter than the photosphere. Consequently, non-thermal energy transport is required to move energy from the energy reservoir of the solar interior past the photosphere before it is deposited as heat in the chromosphere and corona.

Parker (1983) proposed that turbulent convective motions in the near-surface plasma braid the footpoints of coronal magnetic fields, thereby inducing electric currents in the corona; and that the magnetic energy stored in the coronal field due to these currents could then be dissipated to heat the chromosphere and corona. The obligatory cartoon:

Fig. 1: In an "straightened" corona, the magnetic field runs between two photospheric planes, in which footpoints of the coronal fields are anchored. Photospheric convective motions can braid coronal fields, inducing electric currents in the corona. Such currents imply the presence of free magnetic energy in the corona, i.e, energy that can be released by dissipating the currents.

The induced currents might be characterized as either steady or rapidly varying (e.g., wave-driven) relative to the timescales of the atmospheric response, and the dissipation of each has been referred to as "DC" (direct-current) or "AC" (alternating-current) heating (e.g., Ref [3]).

Can we observe the upward transport of magnetic energy that drives heating in the outer solar atmosphere?

The Poynting Flux

The flux of magnetic energy across the photosphere --- in a local Cartesian approximation, the vertical component S_z of the Poynting vector, S --- can be estimated from the observed magnetic evolution in sequentially measured vector magnetograms (2D maps of the photospheric magnetic field vector, B Assuming that the photospheric magnetic field, B, is frozen to the plasma --- a valid assumption in quite general circumstances (see, e.g., Ref. [4]), then the photospheric electric field, E, is given by the ideal Ohm's law,

c E = - (v × B).

Then

S_z = [v_z B_z² - (v_h ⋅ B_h)B_z]/ 4 π ,

where the h subscripts refer to horizontal components of vectors. This expression for total vertical Poynting flux has been conceptually divided into an "emergence" term, which contains v_z, and a "shear" term, which contains v_h (e.g., Ref [5]). In fact, a positive (upward) value for the shearing term also implies the emergence of magnetized plasma across the photosphere. This is because a positive value implies a non-zero component of the photospheric velocity perpendicular to B that is upward. Flows parallel to B can be ignored, since they cancel out in the equation for S_z. Upward transport of a tilted magnetic field does not increase the total unsigned magnetic flux threading the photosphere. (This is in contrast to the emergence of new, or additional, magnetic flux, which must occur at a polarity inversion line).

An Observational Estimate

Plage magnetic fields are unipolar, nearly-vertical fields associated with active regions that are more coherent and typically stronger than quiet-sun fields, but weaker than sunspot (umbral) fields; see, e.g., Ref. [6]. Plage fields are ideal observational targets for estimating energy injection by convection, because they are: (i) strong enough to be measured with relatively small uncertainties; (ii) not so strong that convection is heavily suppressed (as within umbrae); and (iii) unipolar, so S_z in plage is not influenced by flux emergence, which cannot explain steady heating in stable, active-region fields. For observations of B in plage regions that include an estimate of the filling factor f, the upward flux of magnetic energy is approximately given by (Ref. [7])

S_z^plage ≈ - (v_h ⋅ B_h)B_z/ 4 π

Hence, estimating S_z^plage requires both a vector magnetic field measurement and an estimate of v_h. Welsch et al. (2012) used local correlation tracking (LCT) to estimate v_h from a sequence of line-of-sight magnetograms of NOAA AR 10930 observed by the Narrowband Filter Imager (NFI), part of the Solar Optical Telescope (SOT) aboard the joint Japanese/U.S./U.K. Hinode satellite. From simultaneous observations recorded by the SOT's SpectroPolarimeter (SP), B. Lites and the late T.R. Metcalf estimated the vector magnetic field and filling factor. Yeates et al. (2014) and Welsch (2014) combined these results to estimate the Poynting flux in two small regions of plage, and found upward (positive) fluxes of energy. The left panel of Fig. 2 shows a map of B and v_h from the region studied by Welsch (2014), and the right panel shows the corresponding Poynting flux map.

Fig. 2, Left: Grayscale shows filling-factor weighted vertical flux density B_z (black is negative, white positive; saturation at 750 Mx cm^-2), overlain with red vectors showing the horizontal magnetic field B_h and aqua vectors showing the horizontal velocity v_h derived from LCT. Right: Grayscale shows the vertical Poynting flux S_z (black is negative, white positive; saturation at 4 x 10⁸ erg cm^-2 s^-1), derived from the data in Fig. 2. Contours corresponding to -125 Mx cm^-2 and -250 Mx cm^-2 in B_z from Fig. 2 are overlain. Note that both upward and downward energy fluxes are seen, but the net energy flux is upward (positive in our sign convention).

All_fov_plage_mask.png

Mask of plage-like fields in AR 10930, i.e., pixels for which the vector magnetic field was estimated and filling-factor-weighted |Bz| was between 100 and 1500 Mx cm−2 and inclinations were less than 30 from the vertical.

All_fov_distribs_ff.png

Distributions of upward (blue solid) and downward (red dashed) Poynting flux values in plage-like pixels in AR 10930. Upward wins!

The future

The next EUNIS flight, to occur early in 2016, will again observe the Fe XIX 592.2 and Fe XII 592.6 A lines. In addition, the newer version of EUNIS will include a spectral channel to observe the bright line of Fe XVIII at 93.9 A, formed at temperatures near 7.1 million degrees, as well as other lines formed at temperatures between 4 and 10 million degrees. These observations may provide more conclusive support for the nanoflare theory of solar coronal heating. EUNIS is supported by the NASA Heliophysics Division through its "Low Cost Access to Space" program.

References

[1] "Zur Frage der Deutung der Linien im Spektrum der Sonnenkorona"; "Die Deutung der Emissionslinien im Spektrum der Sonnenkorona"

[2] "The Origins of Hot Plasma in the Solar Corona"

[3] "On Solving the Coronal Heating Problem"

[4] "Depth of origin of solar active regions"

[5] "Magnetic Energy and Helicity in Two Emerging Active Regions in the Sun"

[6] "Active Region Magnetic Fields. I. Plage Fields"

[7] "Magnetic Properties at Footpoints of Hot and Cool Loops"