Solar Chromospheric Heating by Magnetohydrodynamic Waves: Dependence on the Inclination of the Magnetic Field

The SOT observations were carried out using an SP (Lites et al. 2013) that recorded four Stokes (I, Q, U, and V) profiles of Fe i lines at 6301.5 and 6302.5 Å with a spectral sampling of 21.55 mÅ. The field of view is 3'' × 82'' with 10 slit positions for a 3'' width at a 03 interval. The Stokes parameters were measured with exposures during the continuous rotation of the polarization modulator in 1.6 s at each slit position, and it takes approximately 21 s to scan the file of view. This is the fast-mapping mode, wherein the exposures were accumulated at two slit positions with a slit width of 015; the two pixels were summed in the slit direction to provide spectral data with a pixel size of 032. In this analysis, we used the level-2 data from the Community Spectropolarimetric Analysis Center at the High Altitude Observatory (HAO)/NCAR. ³ The data was calibrated with the SP_PREP routine in Solar SoftWare (Lites & Ichimoto 2013), followed by an inversion assuming a Milne–Eddington atmosphere with the HAO MERLIN code to obtain physical quantities such as magnetic field strength, azimuth angle, and tilt relative to the line-of-sight direction.

For the magnetic field strength, the line-of-sight component was used in the analysis, and the inclination was analyzed using the inclination of the magnetic field lines from the local zenith of the photosphere. The inclination of the field was transformed from the line-of-sight coordinate system to the local coordinate system. We adopted the angle at which the inclination of the field becomes small relative to the local zenith, due to the 180° ambiguity resolution because the magnetic field lines are expected to continuously tilt from the center of the pore to the periphery.

IRIS observed with the slit (width 0166) for spectroscopy and with the slit jaw imager (SJI), which measures the radiation intensity by imaging observations. The time variation of the Doppler velocity was derived from spectroscopic observations of the Mg ii k line. The spectral sampling is 12.72 mÅ, and the time resolution is 27 s. The slit scanned a 7'' width in eight steps at 1'' intervals. The SJI was acquired every 14 s at four wavelengths: C ii 1330 Å, Si iv 1400 Å, Mg ii k 2796 Å, and Mg ii wing 2830 Å. The SJI enabled the identification of the exact position of the slit. We used the level-2 data created with instrumental calibration, including the dark current subtraction, flat field, and geometrical corrections (De Pontieu et al. 2014).

The wing of the Mg ii k line provides information about the lower part of the chromosphere, whereas the core is sensitive to the upper part of the chromosphere (Leenaarts et al. 2013a, 2013b; Pereira et al. 2013). The Mg ii k line is typically observed in double peaks in the quiet Sun and single peaks in sunspots. The plage may show different shapes, with brighter, wider wings, single peaks, or inverted cores (Morrill et al. 2001). The depth of the core inversion is inversely proportional to the magnetic field strength. This is because the Mg ii k line is a very abundant elemental resonance line; therefore, the line core is optically thick and can form at relatively low densities in non-LTE (Carlsson et al. 2015; Schmit et al. 2015).

The bisector method was used to derive the wavelength of the line center at the intensity level of the line wings; it is the midpoint between λ₁ and λ₂ defined at 40% of the peak intensity of Mg ii k, as shown in Figure 2(a; Graham & Cauzzi 2015). The reference wavelength for the zero Doppler velocity is derived by averaging the line profiles in the spatial and temporal directions for all pixels on each day. The quiet Sun is expected to have an intrinsic velocity of zero or less than a few kilometers per second. However, as we did not obtain data for the quiet Sun in this observation, we assume that the velocity is zero, averaged over the observation region in the spatial and temporal directions. The Doppler velocity was obtained from the deviation of the wavelength from the reference wavelength.

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The wavelength of the line core was defined by detecting three extremes k₁, k₂, and k₃ (k₃ is the minimum value between k₁ and k₂) and applying Gaussian fitting in the range from k₁–k₂ (Figure 2(b)). In the plage region, Mg ii k may be observed as a single peak with a tiny core inverted profile, as shown in Figure 2(c). In such cases, profiles where (intensity decrease at k₃)/(average of the intensities with respect to k₁ and k₂) is less than 15% are considered to be at single peaks, and the wavelength was derived with the Gaussian fit to the whole line profile instead.

The Hinode SOT/SP and IRIS fields of view were aligned to identify the IRIS slit positions on the SP's field of view. The SJI images of Mg ii wing (2832 Å), which are dominantly photospheric, were used for the alignment. Each image was cross-correlated with the SP map taken at the closest time, providing the relative displacement between them.

Figure 3 shows how the SP's and IRIS's fields of view drifted in the x-direction, i.e., east–west direction on the solar surface as a function of the time. At least two IRIS slits are included in the SP's field of view. IRIS slits in the SP's field of view were selected, and SP pixel data overlapping with the selected slits were used for analysis. The positions in the north–south direction were also coaligned in the same manner.

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The Fourier transform was applied to the time profiles to derive the power distribution as a function of the frequency. Figure 7 shows the power spectrum for the three sets of time profiles: (a) in the photosphere derived from the SP data, (b) in the lower chromosphere derived from the wing of the Mg ii k line in IRIS, and (c) in the upper chromosphere derived from the core. The power spectra are the averages of the 5 day data. The inclination of the magnetic field lines θ is classified into 0° < θ < 30°, 30° < θ < 50°, and 50° < θ < 90°. For all data, a subsonic filter (Title et al. 1989; Oba et al. 2017) was applied before the Fourier transfer to remove the frequency component caused by the gas convection at the photosphere. The power distribution below 1.5 mHz was removed because the gas convection is below 2 mHz. Figure 7(a) shows that the power distribution in the photosphere is enhanced at 3 mHz. This may be due to the observation of the 5 minute oscillations (3.3 mHz). Below 5 mHz, there is no difference in the power with respect to the inclination of the magnetic field lines, while above 5 mHz, the power of θ > 30° is slightly larger than that of θ < 30°. As shown in Figure 7(b), in the lower chromosphere, the power of θ > 50° is smaller than the others below 5 mHz, and there is no difference with respect to the inclination of the magnetic field lines above 5 mHz. Figure 7(c) shows that the power of θ > 50° in the upper part of the chromosphere is smaller than the others for all frequencies.

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The energy flux F for the three heights at the photosphere, and lower and upper chromospheres, can be estimated using the following equation (see Sobotka et al. 2016):

$$\begin{eqnarray}&&F={\int }_{{\nu }_{{ac}}}^{{\nu }_{\max }}\rho {P}_{v}(\nu ){v}_{g}(\nu )d\nu ,\end{eqnarray} \tag{ 2 }$$

where ρ is the density, P_v (ν) is the power spectrum of the Doppler velocity obtained from the Fourier transform, and v_g is the group velocity for propagating waves. The group velocity for waves propagating along inclined magnetic field lines is given by the following equation using the cutoff frequency ν_ac obtained from Equation (1) and the speed of sound ${c}_{s}=\sqrt{\gamma p/\rho }$,

$$\begin{eqnarray}&&{v}_{g}={c}_{s}\sqrt{1-{\left({\nu }_{{ac}}/\nu \right)}^{2}}.\end{eqnarray} \tag{ 3 }$$

Here, the acoustic slow wave is essentially field guided (Schunker & Cally 2006), and we therefore multiply the geometric factor $\cos \theta$ in the calculation of the energy flux, in order to derive the energy flux that the acoustic waves bring in the vertical direction upward from the solar surface. The density ρ and temperature T at each height are taken from the atmospheric model VAL-F (Vernazza et al. 1981), which is an atmospheric model for bright internetwork regions.

The energy dissipation at the photosphere was estimated from the difference between the lower chromosphere and the photosphere, while the energy dissipation at the chromosphere was estimated from the difference between the upper and lower chromospheres. The error is obtained from the following equation using the power spectrum error ${\sigma }_{{P}_{v}}$:

$$\begin{eqnarray}&&{\sigma }_{F}=\sqrt{\displaystyle \sum _{v}{\left(\rho {v}_{g}\right)}^{2}{\left({\sigma }_{{P}_{v}}\right)}^{2}}.\end{eqnarray} \tag{ 4 }$$

3.2.1. Dependence on the Inclination of the Magnetic Field

The average amounts of energy dissipated in the photosphere and chromosphere are shown in Figures 8(b) and (a), respectively, as a function of the inclination of the magnetic field. The inclination is measured at the photosphere (Figure 5). The red line represents the amount of energy dissipated by the low-frequency (3–6 mHz) component, while the black line represents the amount of energy dissipated by the high-frequency (6–10 mHz) component. The inclination of the magnetic field lines θ is classified into θ < 30°, 5° each for 30° < θ < 60° and θ > 60°.

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In the photosphere, the low-frequency waves make greater contributions to the energy dissipation. The energy by low-frequency waves increases monotonically with the inclination of the field, with the amount of the energy dissipation exceeding 2 × 10⁵ W m⁻² at the inclination of the field around 50°–60°. The heating contributions of high-frequency waves to the photosphere are almost constant, less dependent on the inclination of the field, and the amount is around (2–6) ×10⁴ W m⁻².

The dependency of the energy dissipation in the chromosphere is different from that observed in the photosphere. The energy dissipated in the chromosphere by the high-frequency waves decreases monotonically as a function of the inclination of the field; 8 × 10³ W m⁻² for an inclination smaller than 30° and 3 × 10³ W m⁻² for an inclination larger than 60°. The energy by low-frequency waves increases with the inclination of the field in the range below 40°, whereas the energy decreases in the range above 40°. The maximum energy dissipation is 9 × 10³ W m⁻² on average at a 40° inclination of the field, which may be slightly lower than the energy required for heating in the chromosphere. Jefferies et al. (2006) demonstrated that low-frequency waves propagate to the upper layers in the region where the magnetic field lines are inclined more than 30°, but Figure 8(a) shows that the energy may not be carried efficiently to the chromospheric height when the magnetic field lines are inclined beyond 40°.

3.2.2. Dependence on the Inclination of Magnetic Field Lines and Field Strength

Figure 9 shows the energy dissipated in the chromosphere and photosphere with respect to the magnetic field strength (line of sight) and inclination. The inclination of the field is in 10° increments on the horizontal axis, and the field strength ∣B∣ is in 100 G increments on the vertical axis. The energy dissipation values are the average of the data in 4 days.

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For the amount of the dissipated energy estimated in the chromosphere, the low-frequency component shows an enhancement when the magnetic field lines with a magnitude higher than 600 G are inclined more than 40° (Figure 9(a)), while no apparent dependence on the inclination angle is observed for the high-frequency component (Figure 9(b)). The low-frequency and high-frequency components show fewer enhancements of the dissipated energy in the weak field pixels, which is different from behaviors in the photosphere, as can be seen in Figure 8(c). The spectral profiles with single or small amplitude peaks (Figure 2(c)) are observed mainly in the region marked by the red box in Figures 9(a) and (b), indicating that they do not considerably affect the overall trend. If we focus on the energy dissipation in the photosphere, the energy dissipation of the low-frequency component is enhanced in the magnetic field below 600 G (Figure 9(c)). The enhanced energy dissipation is also more significant as the inclination of the field increases. For the high-frequency component (Figure 9(d)), no significant enhancements are observed, but the dissipated energy shows weak dependence on the inclination of the field. Also, the amount of energy dissipation is slightly enhanced in the magnetic field below 600 G. This is different from that observed in the low-frequency component.

The maximum magnitude of the energy dissipation by the low-frequency component in the chromosphere is 1.4 ×10⁴ W m⁻² (the value averaged over 4 days) at 40° < θ < 60° and 800 G < ∣B∣ < 1000 G. The dissipated energy may exceed 1 × 10⁴ W m⁻² when the magnetic field with a strength higher than 600 G is inclined by 40° or more from the solar surface. This may satisfy the heating required for the chromosphere. The maximum energy dissipation in the photosphere is 3 × 10⁵ W m⁻² (averaged over 4 days), which is distributed in weak (<400 G) and inclined (>50°) field pixels.

As shown in Figure 9, the energy dissipation of the low-frequency component is more significant in the photosphere when the magnetic field is weaker than 400 G, while the chromosphere shows more energy dissipation by the low-frequency component when the magnetic field lines with a strength higher than 600 G are inclined more than 40° from the solar surface. On the photosphere, a large number of acoustic oscillation modes (p-modes) are mainly excited with periods of about 3–15 minutes (frequencies of about 1–5 mHz), and they are observed as the low-frequency component in weak magnetic field regions. Gas convection, which is the generator of magnetoacoustic waves at the photosphere, becomes weaker in strong magnetic fields, making it harder for waves to be generated. The weak magnetic field region may contain isotropically propagating waves (fast mode of magnetoacoustic waves), which would dissipate before propagating to the chromosphere. As the amount of energy dissipation depends on the inclination of the magnetic field lines in the low-frequency component, the reduced effective gravity on the inclined magnetic field lines makes the cutoff frequency lower, allowing for the low-frequency component of the waves to propagate along the magnetic field lines and to dissipate in the chromosphere. This interpretation suggests that the slow-mode waves propagating along the magnetic field lines contribute significantly to the chromospheric heating. The left side of Figure 10 shows that fast-mode waves propagate isotropically in the weak magnetic field, and the center shows that the high-frequency component can propagate upward into the chromosphere, while the low-frequency component is reflected below the cutoff frequency. The right side shows that the low-frequency component can propagate upward because of the lower cutoff frequency caused by the inclination of the magnetic field lines.

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We have assumed here that the traveling waves generated by the photospheric convection heat the upper atmosphere; however, our analysis cannot provide the phase relations in the temporal profiles of velocities observed at each layer. Thus, we cannot give a definitive conclusion that the amount of energy obtained in this study is due to the dissipation of the traveling waves and their transformation into thermal energy. A part of velocity fluctuations may come from the waves reflected at the upper chromosphere and go downward. There are two reasons why determining the phase relations in velocity fluctuations measured at different heights is difficult: (1) when observing waves propagating along inclined magnetic field lines, the line-of-sight direction in the two measurements may not give the information on the same field line so that the same waves may not be captured at the two layers and (2) the chromosphere is full of plasma flows, making it difficult to distinguish the waves propagating from the photosphere. Figures 7(b) and (c) showed that the power in the magnetic field lines with θ > 50° is smaller than the others. This reduction may be partially explained by the Doppler velocity measurements as the line-of-sight component of waves propagating along the magnetic field lines.

For inclined magnetic field lines, the line-of-sight measurements at two layers may not derive the energy flux on the same magnetic field lines. As a trial, we shifted the observation position in the lower chromosphere (IRIS wing) by 1''–2'' in the east–west direction from the original observation position and compared the power spectrum with that at the photosphere. The result demonstrated that the power spectrum in the lower chromosphere was larger than that in the photosphere for the low-frequency component, depending on the observation position in the lower chromosphere. The data used herein have an extremely narrow field of view to increase the cadence, leading to difficulty in estimating the orientation of the inclined field lines. The 180° ambiguity resolution requires the spatial distribution of the field for a much wider field of view. To investigate the energy dissipation in more detail, tracing the magnetic field lines from the photosphere to the chromosphere is necessary. In the near future, the SUNRISE III experiment, that is, the third flight in series of the SUNRISE balloon-borne stratospheric observatory with a 1 m solar telescope (Barthol et al. 2011), is under planning for re-flight in 2024 and it will provide a high-cadence series of spectropolarimetric data for a much wider field of view and with coverage from the photosphere to the chromosphere. Particularly, the Sunrise chromospheric infrared spectropolarimeter (Katsukawa et al. 2020) on the SUNRISE III experiment will be able to provide some spectral lines seamlessly covering the photosphere through the middle chromosphere (Quintero Noda et al. 2017). Moreover, the focal plane instruments equipped with the ground-based 4 m Daniel K. Inouye Solar Telescope, a facility of the National Solar Observatory, will provide a similar series of spectropolarimetric data.

The gas convection at the photosphere is a source of waves. The turbulent motions excite movement and deformation at the foot of the magnetic field lines. If the opposite polarity flux exists next to the field lines, they may approach each other and cause magnetic reconnection, i.e., nanoflares (Parker 1972), which may also generate another type of wave. Such a wave can contribute to velocity fluctuations. However, imagining that such a wave is included as the dominant source may be difficult because the energy dissipation of the low-frequency component increases with the inclination of the magnetic field lines, not only in the photosphere but also in the chromosphere at least with the inclined magnetic field up to 40°. Our results have been interpreted with slow-mode waves as the most possible mechanism, but velocity fluctuations can also be created by Alfvén waves, which are a candidate for the heating of the corona. However, owing to its incompressible nature, Alfvén waves are unlikely to develop into shock waves in the chromosphere and may contribute little to chromospheric heating.

As described in Section 3.2, we used densities from the VAL-F atmospheric model to derive the energy flux in Figure 8. The VAL-F model is an atmospheric model for the bright internetwork that reproduces an atmosphere closer to the plage we have studied. Here, we discuss at which height the three layers analyzed, i.e., the photosphere, the lower chromosphere, and the upper chromosphere, are located from the solar surface and evaluate how the estimated energy is sensitive to the assumption of density used in the calculation. The SP data is mostly sensitive to approximately 300 km from the solar surface, based on the formation height of the Fe i 6301.5 Å line, and thus, velocities from the SP were considered for the photospheric energy flux.

On the derivation of Doppler velocities at the lower- and upper-chromospheric layers, we used the wing and core profiles in the Mg ii k line, respectively. To evaluate the height where the wing and core portions of the line are formed, we examined a publicly available simulation data cube from a realistic simulation of an enhanced network region (Carlsson et al. 2016). This simulation used the radiation MHD code Bifrost (Gudiksen et al. 2011) to simulate the magnetic structure and dynamics of the outer atmosphere with a magnetic field topology characterized by two dominant opposite polarity islands separated by 8 Mm. The simulated magnetic field topology may be similar to what is observed in our plage regions located between two opposite polarity sunspots, although the spatial scale of the magnetic bipolar structure in the simulation is much smaller than that of our region. Nevertheless, we synthesized spectral profiles of the Mg ii k line in a snapshot from this simulation for an enhanced network region by non-LTE radiative transfer computations with a Rybicky and Hummer radiative transfer code (Uitenbroek 2001) and the method described in detail by Leenaarts et al. (2013b) and Hasegawa (2021). This computation enables us to study the heights where the wing and core of the Mg ii k line are formed. Figure 11 shows the spectral profile averaged over the field of view in a snapshot and the height of the formation at each wavelength position in the central portion of the line. The wing portion for the lower chromosphere is formed at a height of approximately 1000 km, whereas 2000–2500 km for the line core that was fitted for the upper chromosphere. These values are only approximate because they are derived from the averaged spectral profile and are not identical to the actual atmosphere observed herein. As the chromosphere is thinner in the active region than in the quiet region, the density at 2000 km (10,000 K) in VAL-F was assumed for the density in the upper chromosphere. When the formation height of the line core changes from 2000 km (used in the results presented so far) to 1800 and 2400 km in the VAL-F model, such difference in the formation height of the line core provides a very minor change (less than 2%) in the evaluated energy dissipation in the chromosphere.

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