# Solar chromospheric heating by magnetohydrodynamic waves: dependence on
magnetic field inclination

Mayu Koyama  Department of Earth and Planetary Science, The University of
Tokyo,
7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan  Institute of Space and
Astronautical Science, Japan Aerospace Exploration Agency,
3-1-1 Chuo-ku, Sagamihara, Kanagawa 252-5210, Japan  Toshifumi Shimizu
Institute of Space and Astronautical Science, Japan Aerospace Exploration
Agency,
3-1-1 Chuo-ku, Sagamihara, Kanagawa 252-5210, Japan  Department of Earth and
Planetary Science, The University of Tokyo,
7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan

###### Abstract

A proposed mechanism for solar chromospheric heating is that
magnetohydrodynamic waves propagate upward along magnetic field lines and
dissipate their energy in the chromosphere. In particular, compressible
magneto-acoustic waves may contribute to the heating. Theoretically, the
components below the cutoff frequency cannot propagate into the chromosphere;
however, the cutoff frequency depends on the inclination of the magnetic field
lines. In this study, using high temporal cadence spectral data of IRIS and
Hinode SOT spectropolarimeter (SP) in plages, we investigated the dependence
of the low-frequency waves on magnetic-field properties and quantitatively
estimated the amount of energy dissipation in the chromosphere. The following
results were obtained: (a) The amount of energy dissipated by the low-
frequency component (3–6 mHz) increases with the field inclination below 40
degrees, whereas it is decreased as a function of the field inclination above
40 degrees. (b) The amount of the energy is enhanced toward $10^{4}\ \rm
W/m^{2}$, which is the energy required for heating in the chromospheric plage
regions, when the magnetic field is higher than 600 G and inclined more than
40 degree. (c) In the photosphere, the low-frequency component has much more
power in the magnetic field inclined more and weaker than 400 G. The results
suggest that the observed low-frequency components can bring the energy along
the magnetic field lines and that only a specific range of the field
inclination angles and field strength may allow the low-frequency component to
bring the sufficient amount of the energy into the chromosphere.

Sun: photosphere — Sun: chromosphere — Sun: oscillations — Sun: magnetic
fields

##  1 Introduction

The solar chromosphere is the dynamical and thin atmosphere formed above the
solar surface (photosphere); it has a temperature of approximately 10,000 K,
higher than that of the photosphere (6,000 K). The amount of energy required
to retain the chromospheric temperature is $10^{4}\ \rm W/m^{2}$ for active
regions, an order of magnitude larger than the heating for the solar corona,
which is the lower-density atmosphere heated to over 1 MK beyond the magnetic
structures of chromosphere (Withbroe & Noyes, 1977). The physical mechanisms
responsible for transferring the energy to the chromosphere and the corona and
dissipating it there are still in debate. The major candidates are
magnetohydrodynamic (MHD) waves and nanoflares. The turbulent convective gas
motions at the photosphere excite MHD waves propagating to the chromosphere
and the corona along the magnetic field lines (e.g., a review by Jess, 2023,
and references therein), whereas they generate tangential discontinuity and
braiding in magnetic field lines in the corona, leading to numerous tiny
magnetic reconnection events (e.g., a review by Pontin & Hornig, 2020, and
references therein), so-called nanoflares (Parker, 1988). Both mechanisms are
closely associated with dynamical behaviors of the magnetic field lines in the
solar atmosphere.

Compressible magneto-acoustic waves may be evolved to shock waves due to
steepening, and their dissipation may contribute to the heating of the
atmosphere. Therefore, investigating the propagation of magneto-acoustic waves
in the low atmosphere to understand their roles in chromospheric heating is
important. There are two types of magneto-acoustic waves: fast mode—in which
the phase relation between the gas pressure and magnetic pressure is in-
phase—and slow mode—in which the phase relation between the gas pressure and
magnetic pressure are in the opposite phase. The slow mode propagates along
magnetic field lines. In the presence of a magnetic field, sunspots are the
most apparent regions for typical display of oscillations in intensity and
Doppler velocity. The power spectra are dominated for 5 min (approximately 3
mHz) in the photosphere and 3 min (approximately 5 mHz or higher) in the
chromosphere (Centeno et al., 2009). The oscillations in the umbrae and
penumbrae of sunspots have been generally interpreted as MHD waves. Kanoh et
al. (2016) conducted simultaneous observations of the Hinode and IRIS
satellites and showed the existence of the slow mode waves in a sunspot umbra
based on the phase relation of waves observed at the photosphere and the
chromosphere. In addition, the energy flux in each layer was derived and the
energy dissipation was estimated from the difference. They obtained an energy
flux of $8.3\times 10\ \rm W/m^{2}$ for waves in the upper chromosphere and
$2.0\times 10^{4}\ \rm W/m^{2}$ for the photosphere, demonstrating that the
difference is sufficient to heat the chromosphere. Oscillating signatures in
sunspot penumbrae, known as running penumbral waves, are interpreted as upward
propagating slow-model waves guided by the magnetic field lines (Löhner-
Böttcher & Bello González, 2015). The dominant frequency of the waves changes
toward lower values as one moves from the umbral center to the surroundings
where the magnetic fields are more inclined to the surface (Bloomfield et al.,
2007; Jess et al., 2013).

The critical frequency at which a wave can propagate is called the cutoff
frequency; it is defined as $\nu_{ac}=\frac{\gamma g}{4\pi c_{s}}$, where
$c_{s}$ is the sound speed, $\gamma$ is the specific heat ratio of a monatomic
molecule, and $g$ is the gravitational acceleration (Priest 2014). The cutoff
frequency is at its maximum in the lowest temperature layer, i.e., the
temperature minimum, clearly shown as the height variation of the cutoff
frequency in observations of a sunspot umbra (Felipe et al., 2018) and in
numerical models (Felipe & Sangeetha, 2020). The low-frequency waves below the
frequency cannot propagate into the chromosphere. In the quiet Sun, where the
plasma $\beta$ is much larger than 1, the cutoff frequency is approximately
5.2 mHz ($\gamma=5/3,\ g=274\rm m/s,\it c_{s}=\rm 7\ km/s$). Bel & Leroy
(1977) showed theoretically that the cutoff frequency depends on the structure
of the magnetic field. In the region of strong magnetic field ($\beta<1$), the
cutoff frequency depends on the inclination angle of the magnetic field lines
with respect to the solar surface. This is because the effective gravity on
the magnetic field lines changes, and the cutoff frequency becomes

$\nu_{ac}=\frac{\gamma g\cos\theta}{4\pi c_{s}}.$ (1)

Note that the cutoff frequency given in equation (1) is only applicable in an
isothermal atmosphere and different forms of the cut-off frequency are
inferred from various representations of the wave equation depending on choice
of variables in more general atmospheres (Schmitz & Fleck, 1998). McIntosh &
Jefferies (2006) observed that low-frequency magneto-acoustic waves below the
cutoff frequency (5.2 mHz) propagate into the chromosphere in the sunspot
penumbra where the magnetic field is highly tilted to the surface. Numerical
simulations of wave propagation in the low atmosphere also show that the
magnetic field inclination is crucial for the propagation of low-frequency
waves through dynamic magnetic structures (Heggland et al., 2011) and wave
energy flux (Schunker & Cally, 2006; Cally & Moradi, 2013).

For observationally investigating the significance of wave contributions to
the upper atmosphere, quantitatively estimating the amount of energy magneto-
acoustic waves transported from the photosphere to the chromosphere and
deposited in the chromosphere considering the magnetic field environment is
essential. However, the number of studies on magnetic plages outside sunspots
is largely limited. Sobotka et al. (2016) used the Interferometric
Bidimensional Spectrometer (IBIS; Cavallini, 2006) at the Dunn Solar Telescope
(DST) to evaluate the deposited acoustic energy flux from the power spectra of
Doppler oscillations measured in the $\rm Ca\ II$ 853.2 nm line core in
comparison to the radiative loss. In active areas around sunspots, the amount
of energy dissipated by 4–9 mHz waves was estimated to be up to $\rm 5\times
10^{3}\ W/m^{2}$ in the height range 1000–1500 km from the solar surface (the
middle chromosphere). The estimated flux is slightly insufficient to heat the
chromosphere; however, the heating contribution was revealed to increase from
23 % in chromosphere network to 54 % in a plage. Abbasvand et al. (2020a) used
observations of the $\rm Ca\ II$ 854.2 nm, H$\alpha$ and H$\beta$ lines with
the Fast Imaging Solar Spectrograph (FISS; Chae et al., 2013) at the 1.6-m
Goode Solar Telescope and the echelle spectrograph attached to the German
Vacuum Tower Telescope (von der Lühe, 1998) and derived that the acoustic
energy flux at frequencies up to 20 mHz deposited in the heights from 1000 to
1400 km can be balanced by the radiative loss in a quiet region, but the
deposited acoustic flux is insufficient in the upper chromosphere higher than
1400 km in both quiet and plage regions. Abbasvand et al. (2020b) used
observations of the $\rm Ca\ II$ 854.2-nm line with the IBIS and derived the
acoustic energy flux in the plage, demonstrating that it contributes by
$50\%\\--60$% in locations with vertical magnetic field and $70\%\\--90$% in
regions where the magnetic field is inclined more than $50^{\circ}$ to the
solar surface normal.

In this study, we investigate roles of magnetic field properties in
oscillations observed in plages, which are observed brightly in the
chromosphere; consequently, heating is inferred to be relatively high so that
we could understand more details of energy transport and dissipation by waves.
For quantitative understanding, we estimate the energy dissipation in the
chromosphere by comparing the energy flux at three altitudes in the plages:
the photosphere, lower chromosphere, and upper chromosphere. In addition, as
the cutoff frequency varies depending on the magnetic-field structure of the
photosphere, we estimate the energy dissipation in the chromosphere including
the energy flux due to low-frequency waves (less than 5 mHz) below the cutoff
frequency. Determining the inclination of the magnetic field lines with
respect to the solar surface with the Hinode SOT/SP, we verify whether low-
frequency waves below the cutoff frequency propagate into the chromosphere
depending on the inclination of the magnetic field lines. We describe
observational methodologies in Section 2. Section 3 shows the observational
results, and Section 4 discusses the interpretation of the observation
results. A summary of this paper is given in Section 5.

##  2 Observations and data analysis

Simultaneous observations of the plage region were performed by the Solar
Optical Telescope (SOT) (Tsuneta et al., 2008; Suematsu et al., 2008; Shimizu
et al., 2008; Ichimoto et al., 2008) onboard the Hinode satellite (Kosugi et
al. 2007) and the IRIS satellite (De Pontieu et al. 2014) for approximately
one hour each day on February 9–12, 2018. The observations were carried out
along with the observation proposal named IRIS-Hinode Operation Plan (IHOP)
0341. The period of each simultaneous Hinode and IRIS observation and the
heliocentric coordinate of the center of the leftmost IRIS slit are shown in
Table 1. Each observation region covers a plage region between the leading and
following sunspots in active region NOAA 12699. The observation area on
February 10, 2018, as an example, is shown in Figure 1. The blue frame shows
the SOT field of view, whereas the red lines show the slit positions of IRIS.

Table 1: Coordinated observation period and coordinate Date | Time (UT) | Position $(x,y)$
---|---|---
09 February 2018 | 12:59:10 – 14:09:58 | $(-335^{\prime\prime},-17^{\prime\prime})$
10 February 2018 | 11:35:13 – 12:44:33 | $(-122^{\prime\prime},-9^{\prime\prime})$
11 February 2018 | 12:05:11 – 13:09:32 | $(104^{\prime\prime},-4^{\prime\prime})$
12 February 2018 | 01:05:11 – 02:14:32 | $(226^{\prime\prime},-5^{\prime\prime})$
Figure 1: Observation area on 10 February 2018. Blue frame shows the field of
view captured by the Hinode SOT observation, whereas red lines show the slit
positions for the IRIS observation. The IRIS slit positions are 8 steps at
$1^{\prime\prime}$ intervals. The SOT field of view contains 10 slit positions
of the spectro-polarimeter measurement with $0.3"$ interval.

###  2.1 Hinode SOT observation and analysis

The SOT observations were carried out by a spectro-polarimeter (SP) (Lites et
al. 2013) that recorded four Stokes $\rm(I,Q,U,V)$ profiles of $\rm Fe\ I$
lines at $\rm 6301.5\AA$ and $\rm 6302.5\AA$ with a spectral sampling of $\rm
21.55\ m\AA$. The field of view is $3^{\prime\prime}\times 82^{\prime\prime}$
with 10 slit positions for a $3^{\prime\prime}$ width at $0.3^{\prime\prime}$
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 $0.15^{\prime\prime}$ ; the two pixels were
summed in the slit direction to provide spectral data with a pixel size of
$0.32^{\prime\prime}$. In this analysis, we used the level-2 data from the
Community Spectro-polarimetric Analysis Center (CSAC) at HAO/NCAR
111https://www2.hao.ucar.edu/csac. The data was calibrated with the SP_PREP
routine in Solar SoftWare (SSW) (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 field
inclination was transformed from the line-of-sight coordinate system to the
local coordinate system. We adopted the angle at which the field inclination
becomes small relative to the local zenith due to the $180^{\circ}$ ambiguity
resolution because the magnetic field lines are expected to continuously tilt
from the center of the pore to the periphery.

###  2.2 IRIS observation and analysis

IRIS was observed with the slit (width $0.166^{\prime\prime}$) 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 $\rm Mg\ II\ k$ line. The
spectral sampling is $12.72\rm m\AA$, and the time resolution is 27 s. The
slit scanned a $7^{\prime\prime}$ width in 8 steps at $1^{\prime\prime}$
intervals. The SJI was acquired every 14 s at four wavelengths: $\rm C\ II\
1330\AA$, $\rm Si\ IV\ 1400\AA$, $\rm Mg\ II\ k\ 2796\AA$, and $\rm Mg\ II\
wing\ 2830\AA$. The SJI enabled the identification of the exact position of
the slit. We used the level-2 data created with the instrumental calibration
including the dark current subtraction, flat field, and geometrical
corrections (De Pontieu et al. 2014).

The wing of the $\rm Mg\ II\ k$ 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, b, Pereira et al. 2013). The $\rm Mg\
II\ k$ 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 $\rm Mg\ II\ k$ line is a very abundant elemental resonance
line; therefore, the line core is optically thick and can form at relatively
low densities in nonlocal thermodynamic equilibrium (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 of $\lambda_{1}$ and
$\lambda_{2}$ defined at 40% of the peak intensity of $\rm Mg\ II\ k$, as
shown in Figure 2(a) (Graham & Cauzzi 2015). The reference wavelength for the
zero Doppler velocity is derived by averaging the line profiles averaged in
the spatial and temporal directions for all pixels on each day. The quiet Sun
is expected to have intrinsic velocity of zero or less than a few $\rm km/s$.
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.

The wavelength of the line core was defined by detecting three extremes
$k_{1},k_{2},k_{3}$ ($k_{3}$ is the minimum value between $k_{1}$ and $k_{2}$)
and applying Gaussian fitting in the range from $k_{1}$ to $k_{2}$ (Figure
2(b)). In the plage region, $\rm Mg\ II\ k$ 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 $k_{3}$)/(average of the intensities
with respect to $k_{1}$ and $k_{2}$) 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.

(a) wing

(b) core

(c) core (single peak)

Figure 2: Analysis for $\rm Mg\ II\ k$ to derive line-of-sight velocity. (a)
The bisector at 40% of the peak intensity for velocity at the lower
chromosphere. (b) The Gaussian fitting to a core inverted profile for velocity
at the upper chromosphere. (c) The Gaussian fitting to the whole line profile
is used instead if the core inverted profile is tiny.

###  2.3 Co-alignment of spectral lines

The Hinode SOT/SP and IRIS fields of view were aligned to identify the IRIS
slit positions on the SP field of view. The SJI images of $\rm Mg\ II\ wing\
(2832\AA)$, 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 and IRIS fields of view were drifted in the X, i.e.,
E-W direction on the solar surface as a function of the time. At least two
IRIS slits are included in SP field of view. IRIS slits in the SP field-of-
view were selected, and SP pixel data overlapping with the selected slits were
used for analysis. The positions in the N-S direction were also co-aligned in
the same manner.

Figure 3: SP and IRIS fields of view in the x-direction as a function of time.
Black line: the slit position of the three IRIS slits located within the SP
field of view. Blue dotted lines: the ends of SP field of view. Red line: the
center of SP field of view. The figure is for the data captured on February
10, 2018.

##  3 Results

The line-of-sight magnetic field strength and the inclination of the magnetic
field in the observation regions are shown in Figures 4 and 5. The inclination
of the magnetic field, denoted by $\theta$, is defined in the local frame
coordinate, i.e., $0^{\circ}$ in the normal direction of the solar surface and
$90^{\circ}$ in the horizontal direction. From the negative polarity of
$y=20^{\prime\prime}$ – $25^{\prime\prime}$ in Figure 5, the magnetic field
lines are continuously inclined from the center of the pore (a small sunspot
with no penumbra), where the magnetic field strength is relatively strong, to
the surroundings. Pixels with the degree of polarization of less than 1%,
which are considered unsuitable for Stokes profile inversion with good
accuracy, were not used in the analysis.

Figure 6 shows an example of the time series of the Doppler velocities
observed in a pixel on February 10, 2018. The pixel has a weak magnetic field
strength (353 G) and tilted magnetic field lines ($\theta=54^{\circ}$). The
amplitudes in each layer are observed to be within 0.8 km/s in the
photosphere, approximately 2 km/s in the lower chromosphere, and up to 10 km/s
in the upper chromosphere. The amplitude of the waves is larger in the upper
layers. This is because the density decreases in the upper layers. We derived
the power distribution by Fourier transforming each wave because determining
the phase difference from the waves observed in each layer is difficult.

Figure 4:  LOS magnetic field strength. The unit on the vertical axis is
arcsec. Figure 5: Inclination of magnetic field lines. The unit on the
vertical axis is arcsec.

(a) In the photosphere derived from the SP data

(b) In the lower chromosphere derived from IRIS $\rm Mg\ II\ k$ wing

(c) In the upper chromosphere derived from IRIS $\rm Mg\ II\ k$ core

Figure 6: Time series of the Doppler velocity observed in a pixel region on
February 10, 2018.

###  3.1 Dependence of power spectrum on inclination of magnetic field lines

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 $\rm Mg\
II\ k$ line in IRIS, and (c) in the upper chromosphere derived from the core.
The power spectra are the averages of the five-day data. The inclination of
the magnetic field lines $\theta$ is classified into
$0^{\circ}<\theta<30^{\circ}$, $30^{\circ}<\theta<50^{\circ}$, and
$50^{\circ}<\theta<90^{\circ}$. 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 photosphere is enhanced
at 3 mHz. This may be due to the observation of the 5-min 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
$\theta>30^{\circ}$ is slightly larger than that of $\theta<30^{\circ}$. As
shown in Figure 7(b), in the lower chromosphere, the power of
$\theta>50^{\circ}$ 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 $\theta>50^{\circ}$ in the upper part
of the chromosphere is smaller than the others for all frequencies.

(a) In the photosphere derived from the SP data

(b) In the lower chromosphere derived from IRIS $\rm Mg\ II\ k$ wing

(c) In the upper chromosphere derived from IRIS $\rm Mg\ II\ k$ core

Figure 7: Power distribution as a function of the frequency. The inclination
of the magnetic field lines; blue lines show $0^{\circ}<\theta<30^{\circ}$,
black lines show $30^{\circ}<\theta<50^{\circ}$, red lines show
$50^{\circ}<\theta<90^{\circ}$.

###  3.2 Energy dissipation in the photosphere and chromosphere

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

$F=\int_{\nu_{ac}}^{\nu_{max}}\rho P_{v}(\nu)v_{g}(\nu)d\nu$ (2)

where $\rho$ is the density, $P_{v}(\nu)$ 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 $\nu_{ac}$ obtained from Equation (1) and the sound speed
$c_{s}=\sqrt{\gamma p/\rho}$,

$v_{g}=c_{s}\sqrt{1-(\nu_{ac}/\nu)^{2}}.$ (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 which the
acoustic waves bring in the vertical direction upward from the solar surface.
The density $\rho$ 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 inter-network 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 $\sigma_{P_{v}}$.

$\sigma_{F}=\sqrt{\sum_{v}(\rho v_{g})^{2}(\sigma_{P_{v}})^{2}}$ (4)

####  3.2.1 Dependence on the inclination of magnetic field

The average amounts of energy dissipated in the photosphere and chromosphere
are shown in Figure 8(b) and Figure 8(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 low frequency (3–6 mHz) component, while the black line
represents the amount of energy dissipated by high frequency (6–10 mHz)
component. The inclination of the magnetic field lines $\theta$ is classified
into $\theta<30^{\circ}$, $5^{\circ}$ each for $30^{\circ}<\theta<60^{\circ}$,
and $\theta>60^{\circ}$.

Figure 8:  The average amount of energy dissipated (a) in the chromosphere and
(b) in the photosphere as a function of the inclination of the magnetic field
lines. Red line: low frequency (3–6 mHz), Black line: high frequency (6–10
mHz).

In the photosphere, the low-frequency waves make more contributions to the
energy dissipation. The energy by low-frequency waves increases monotonically
with the field inclination, reaching the amount of the energy dissipation
exceeding $2\times 10^{5}$ W/m2 at the field inclination around
$50^{\circ}-60^{\circ}$. The heating contributions of high-frequency waves to
the photosphere are almost constant, less dependent of the field inclination,
and the amount is around $(2-6)\times 10^{4}$ W/m2.

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 field
inclination; $8\times 10^{3}$ W/m2 in the inclination smaller than
$30^{\circ}$ and $3\times 10^{3}$ W/m2 in the inclination larger than
$60^{\circ}$. The energy by low-frequency waves increases with the field
inclination in the range below $40^{\circ}$, whereas the energy decreases in
the range above $40^{\circ}$. The maximum energy dissipation is $9\times
10^{3}$ W/m-2 on average at the field inclination of $40^{\circ}$, 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 $30^{\circ}$, but Figure 8(a) shows that the energy may not be carried
efficiently to the chromospheric height when the magnetic field lines are more
inclined beyond $40^{\circ}$.

####  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
field inclination is in $10^{\circ}$ increments in the horizontal axis, and
the field strength $|B|$ is in $\rm 100\ G$ increments in the vertical axis.
The energy dissipation values are the average of the data in four days.

Figure 9:  Amount of the dissipated energy as a function of the inclination of
the magnetic field lines in the horizontal axis and the magnetic field
strength in the vertical axis. (a) and (b) are the energy dissipated in the
chromosphere by the low-frequency component (3-6 mHz) and the high-frequency
component (6-10 mHz), respectively. (c) and (d) are the same but for the
photosphere. The red box in (a) and (b) gives the condition where the line
profile of $\rm Mg\ II\ k$ line is a single peak with only a tiny core
inverted profile. The velocity for the upper chromosphere was derived by a
whole profile fitting instead of the fitting to the core inverted profile as
described in section 2.2. The values are acquired by averaging the data for
four days.

For the amount of the dissipated energy estimated in the chromosphere, the
low-frequency component shows an enhancement when the magnetic field lines
with the magnitude higher than 600 G are inclined more than $40^{\circ}$
(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 less enhancements of the dissipated energy in
the weak field pixels, which is different from behaviors in the photosphere as
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 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 field inclination increases. For the high-frequency
component (Figure 9(d)), no significant enhancements are observed but the
dissipated energy shows weak dependence on the field inclination, Also, the
amount of energy dissipation is slightly enhanced in the magnetic field below
600 G. It 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 $\rm 1.4\times 10^{4}\ W/m^{2}$ (The value averaged
over four days) at $40^{\circ}<\theta<60^{\circ}$ and $\rm 800~{}G<|\it
B\rm|<1,000~{}G$. The dissipated energy may exceed $\rm 1\times 10^{4}\
W/m^{2}$ when the magnetic field with the strength higher than 600 G is
inclined by 40 degree or more from the solar surface. This may satisfy the
heating required for the chromosphere. The maximum energy dissipation in the
photosphere is $\rm 3\times 10^{5}\ W/m^{2}$ (averaged over four days), which
are distributed in weak ($<400$ G) and inclined ($>50^{\circ}$) field pixels.

##  4 Discussions

###  4.1 Interpretations of the results

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 the strength higher than 600 G
are inclined more than $40^{\circ}$ 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 min (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 the 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 high-frequency component can propagate upward into the
chromosphere while low-frequency component is reflected below the cutoff
frequency. The right side shows that low-frequency component can propagate
upward because of lower cutoff frequency caused by the inclination of the
magnetic field lines.

Figure 10: Waves inferred from results (schematic)

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 downwards.
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; (2)
the chromosphere is full of plasma flows, making it difficult to distinguish
the waves propagating from the photosphere. Figure 7(b) and (c) showed that
the power in the magnetic field lines with $\theta>50^{\circ}$ 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 $1^{\prime\prime}$ – $2^{\prime\prime}$ 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-degree 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-3 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 high-cadence series of spectro-polarimetric data for much wider field
of view and with coverage from the photosphere to the chromosphere.
Particularly, the Sunrise Chromospheric Infrared spectroPolarimeter (SCIP;
Katsukawa et al., 2020) on the SUNRISE-3 experiment will be able to provide
some spectral lines seamlessly covering from the photosphere through the
middle chromosphere (Quintero Noda et al., 2017). Moreover, the focal plane
instruments equipped to the ground-based 4-m Daniel K. Inouye Solar Telescope
(DKIST), a facility of the National Solar Observatory (NSO), will provide
similar series of spectro-polarimetric 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 waves. 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 degree. 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 the
incompressible nature, Alfvén waves are unlikely to develop to shock waves in
the chromosphere and may contribute little to the chromospheric heating.

###  4.2 Formation height and density’s assumption

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 inter-network 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 $\rm Fe\ I\ 6301.5\ \AA$
line, and thus, velocities from the SP were considered for the photospheric
energy flux.

Figure 11: Spectral profile of the $\rm Mg\ II\ k$ line averaged over the
field of view (black), and the height of the formation at each wavelength
position of the line (red).

On the derivation of Doppler velocities at the lower and upper chromospheric
layers, we used the wing and core profiles in the $\rm Mg\ II\ k$ 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 magnetohydrodynamics 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 $\rm Mg\ II\ k$
line in a snapshot from this simulation for an enhanced network region by non-
LTE radiative transfer computations with RH 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 $\rm Mg\ II\ k$ line are formed. Figure 11 shows the spectral profile
averaged over the field of view in a snapshot and its 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 the height approximately 1,000
km, whereas 2,000–2,500 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 2,000 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 2,000 km (used in the results presented so far) to 1,800 km
and 2,400 km in the VAL-F model, such difference in the formation height of
the line core provides a very minor change (less than 2 percent) in the
evaluated energy dissipation in the chromosphere.

##  5 Summary

In this study, we quantitatively estimated the magnetic field structure and
derived how the energy dissipations at the photosphere and chromosphere depend
on the properties of magnetic fields in the plage using the simultaneous
observations by Hinode and IRIS. The results show that the energy dissipation
by low-frequency waves at 3–6 mHz increases at the chromosphere in the range
of the magnetic field inclination below 40 degree, and it begins to decrease
when the field is more inclined beyond 40 degree. This is considered to be due
to the propagation of low-frequency waves into the chromosphere. The effective
gravity on the magnetic field lines decreases due to the tilt of the magnetic
field lines, resulting in decreasing cutoff frequency. We interpreted that the
waves observed in the chromosphere are mostly slow-mode waves propagating
along the magnetic field lines. The low-frequency waves may be able to bring
the energy in the vertical direction, although too highly inclined field may
suppress the transport of energy in the vertical direction. The plage is
observed to be bright in the chromosphere and is considered to be important
for chromospheric heating. The previous studies, however, have not
quantitatively described the energy dissipation above the plage as far as
their authors know. Few studies have quantitatively utilized the magnetic
field structure derived in the photosphere. The energy flux obtained by
previous studies is less than the heating rate required for the chromosphere
(Sobotka et al. 2016). In this study, we found that the low-frequency
component in 3–6 mHz can dissipate more than $10^{4}\ \rm W/m^{2}$ of energy
in the chromosphere in only a specific configuration of the magnetic field
lines. Furthermore, the amount of energy dissipated in the photosphere
increases with decreasing magnetic field strength at the photosphere, whereas
the amount of energy dissipated in the chromosphere is more significant in the
magnetic field strength higher than 600 G.

The authors thank the anonymous referee for pointing out a flaw in the energy
assessment in draft manuscript. Hinode is a Japanese mission developed and
launched by ISAS/JAXA, with NAOJ as domestic partner and NASA and STEC (UK) as
international partners. It is operated by these agencies in co-operation with
ESA and NSC (Norway). IRIS is a NASA small explorer mission developed and
operated by LMSAL with mission operations executed at NASA Ames Research
Center and major contributions to downlink communications funded by ESA and
the Norwegian Space Centre. We sincerely thank the Hinode team and the IRIS
team for providing the coordinated observations used in this article. This
work was partially supported by JSPS KAKENHI Grant No. JP18H05234.

## References

  * Abbasvand et al. (2020a) Abbasvand, V., Sobotka, M., Heinzel, P., et al. 2020a, ApJ, 890, 22
  * Abbasvand et al. (2020b) Abbasvand, V., Sobotka, M., Švanda, M., et al. 2020b, A&A, 642, A52
  * Barthol et al. (2011) Barthol, P., Gandorfer, A., Solanki, S. K., et al. 2011, Sol. Phys., 268, 1
  * Bel & Leroy (1977) Bel, N., & Leroy, B. 1977, A&A, 55, 239
  * Bloomfield et al. (2007) Bloomfield, D., Lagg, A., & Solanki, S. 2007, ApJ, 671, 1005
  * Cally & Moradi (2013) Cally, P., & Moradi, H. 2013, MNRAS, 435, 2589
  * Carlsson et al. (2016) Carlsson, M., Hansteen, V. H., Gudiksen, B. V., Leenaarts, J., & De Pontieu, B. 2016, A&A, 585, A4
  * Carlsson et al. (2015) Carlsson, M., Leenaarts, J., & De Pontieu, B. 2015, ApJL, 809, L30
  * Cavallini (2006) Cavallini, F. 2006, Sol. Phys., 236, 415
  * Centeno et al. (2009) Centeno, R., Collados, M., & Trujillo Bueno, J. 2009, ApJ, 692, 1211
  * Chae et al. (2013) Chae, J., Park, H.-M., & Ahn, K. e. a. 2013, Solar Physics, 288, 1
  * De Pontieu et al. (2014) De Pontieu, B., Rouppe van der Voort, L., Pereira, T. M. D., et al. 2014, in American Astronomical Society Meeting Abstracts, Vol. 224, American Astronomical Society Meeting Abstracts #224, 313.02
  * Felipe et al. (2018) Felipe, T., Kuckein, C., & Thaler, I. 2018, A&A, 617, A39
  * Felipe & Sangeetha (2020) Felipe, T., & Sangeetha, C. R. 2020, A&A, 640, A4
  * Graham & Cauzzi (2015) Graham, D. R., & Cauzzi, G. 2015, APJL, 807, L22
  * Gudiksen et al. (2011) Gudiksen, B. V., Carlsson, M., Hansteen, V. H., et al. 2011, A&AP, 531, A154
  * Hasegawa (2021) Hasegawa, T. 2021, PhD thesis, The University of Tokyo
  * Heggland et al. (2011) Heggland, L., Hansteen, V. H., De Pontieu, B., & Carlsson, M. 2011, ApJ, 743, 142
  * Ichimoto et al. (2008) Ichimoto, K., Lites, B., Elmore, D., et al. 2008, Solar Physics, 249, 233
  * Jefferies et al. (2006) Jefferies, S. M., McIntosh, S. W., Armstrong, J. D., et al. 2006, ApJL, 648, L151
  * Jess (2023) Jess, D. B. 2023, Living Reviews in Solar Physics, 20, 1
  * Jess et al. (2013) Jess, D. B., E., R. V., van Doorsselaere, T., Keys, P. H., & Mackay, D. H. 2013, ApJ, 779, 168
  * Kanoh et al. (2016) Kanoh, R., Shimizu, T., & Imada, S. 2016, ApJ, 831, 24
  * Katsukawa et al. (2020) Katsukawa, Y., del Toro Iniesta, J. C., Solanki, S. K., et al. 2020, in Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 11447, Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, 114470Y
  * Kosugi et al. (2007) Kosugi, T., Matsuzaki, K., Sakao, T., et al. 2007, Sol. Phys., 243, 3
  * Leenaarts et al. (2013a) Leenaarts, J., Pereira, T. M. D., Carlsson, M., Uitenbroek, H., & De Pontieu, B. 2013a, ApJ, 772, 89
  * Leenaarts et al. (2013b) —. 2013b, ApJ, 772, 90
  * Lites & Ichimoto (2013) Lites, B. W., & Ichimoto, K. 2013, Solar Physics, 283, 601
  * Lites et al. (2013) Lites, B. W., Akin, D. L., Card, G., et al. 2013, Sol. Phys., 283, 579
  * Löhner-Böttcher & Bello González (2015) Löhner-Böttcher, J., & Bello González, N. 2015, A&A, 580, A53
  * McIntosh & Jefferies (2006) McIntosh, S. W., & Jefferies, S. M. 2006, ApJL, 647, L77
  * Morrill et al. (2001) Morrill, J. S., Dere, K. P., & Korendyke, C. M. 2001, ApJ, 557, 854
  * Oba et al. (2017) Oba, T., Iida, Y., & Shimizu, T. 2017, ApJ, 836, 40
  * Parker (1972) Parker, E. N. 1972, ApJ, 174, 499
  * Parker (1988) —. 1988, ApJ, 330, 474
  * Pereira et al. (2013) Pereira, T. M. D., Leenaarts, J., De Pontieu, B., Carlsson, M., & Uitenbroek, H. 2013, ApJ, 778, 143
  * Pontin & Hornig (2020) Pontin, D., & Hornig, G. 2020, Living Reviews in Solar Physics, 17, 5
  * Priest (2014) Priest, E. 2014, Magnetohydrodynamics of the Sun (Cambridge University Press), doi:https://doi.org/10.1017/CBO9781139020732
  * Quintero Noda et al. (2017) Quintero Noda, C., Shimizu, T., & Katsukawa, Y. e. a. 2017, MNRAS, 464, 4534
  * Schmit et al. (2015) Schmit, D., Bryans, P., De Pontieu, B., et al. 2015, ApJ, 811, 127
  * Schmitz & Fleck (1998) Schmitz, F., & Fleck, B. 1998, A&A, 337, 487
  * Schunker & Cally (2006) Schunker, H., & Cally, P. 2006, MNRAS, 372, 551
  * Shimizu et al. (2008) Shimizu, T., Nagata, S., Tsuneta, S., et al. 2008, Solar Physics, 249, 221
  * Sobotka et al. (2016) Sobotka, M., Heinzel, P., Švanda, M., et al. 2016, ApJ, 826, 49
  * Suematsu et al. (2008) Suematsu, Y., Tsuneta, S., Ichimoto, K., et al. 2008, Solar Physics, 249, 197
  * Title et al. (1989) Title, A. M., Tarbell, T. D., Topka, K. P., et al. 1989, ApJ, 336, 475
  * Tsuneta et al. (2008) Tsuneta, S., Ichimoto, K., Katsukawa, Y., et al. 2008, Sol. Phys., 249, 167
  * Uitenbroek (2001) Uitenbroek, H. 2001, ApJ, 557, 389
  * Vernazza et al. (1981) Vernazza, J. E., Avrett, E. H., & Loeser, R. 1981, ApJS, 45, 635
  * von der Lühe (1998) von der Lühe, O. 1998, New Astron. Rev., 42, 493
  * Withbroe & Noyes (1977) Withbroe, G. L., & Noyes, R. W. 1977, ARA&A, 15, 363