with the cross-hatched/shaded regions. The arrows on the right panels show the
phase angles between oscillations at the two atmospheric heights, with in-
phase oscillations depicted by arrows pointing right and fluctuations in Ca ii
H leading those in 300 nm by 90 degrees marked by arrows pointing straight up.
Images reproduced from Jafarzadeh et al. (2017d).

As such, the global wavelet spectrum is often considered a biased estimation
of the true Fourier spectrum (Wu and Liu, 2005). This effect can be clearly
seen in Figure 24, which displays both the Fourier and global wavelet power
spectra for the same HARDcam H$\alpha$ time series shown in the lower panel of
Figure 6. In Figure 24, the higher power at larger scales (lower frequencies)
is visible in the global wavelet spectrum (red line), when compared to that
derived through traditional Fourier techniques (black line). However, at
smaller scales (higher frequencies), both the global wavelet and Fourier
spectra are in close agreement with one another, with the global wavelet
spectrum appearing as a smoothed Fourier spectrum. The reason for these
effects is due to the width of the wavelet filter in Fourier space. At large
scales (low frequencies), the wavelet is narrower in frequency, resulting in
sharper peaks that have inherently larger amplitudes. Contrarily, at small
scales (high frequencies), the wavelet is more broad in frequency, hence
causing any peaks in the spectrum to become smoothed (Torrence and Compo,
1998). As such, it is important to take such biases into consideration when
interpreting any embedded wave motion. Indeed, Banerjee et al. (2001),
Christopoulou et al. (2003), Samanta et al. (2016), Kayshap et al. (2018), and
Chae et al. (2019) have discussed the implementation of global wavelet and
Fourier power spectra in the context of solar oscillations.

####  2.4.1 Wavelet Phase Measurements

Similar to the Fourier phase lag analysis described in Section 2.2.5, it is
also useful to obtain phase angles, cross-power spectrum, and coherence
between wavelet power spectra at different wavelengths, spatial locations,
and/or multi-component spectral measurements. Hence, the phase angles are
determined not only as a function of frequency, but also as a function of
time. These phase angles are usually demonstrated as small arrows on a wavelet
co-spectrum (or wavelet coherence) map, where their directions indicate the
phase angles at different time-frequency locations. The convention with which
an arrow direction represents, e.g., zero and 90 degrees phase angles (and
which lightcurve leads or lags behind) should be specified.

Figure 26: Phase diagram of transverse oscillations in 7 small magnetic
elements observed in two layers of the lower solar atmosphere (with $\approx
450$ km height difference) from SuFI/Sunrise. Image reproduced from Jafarzadeh
et al. (2017d).

Reproduced from Jafarzadeh et al. (2017d), the lower- and upper-left panels of
Figure 25 display two wavelet power spectra (from a Morlet wavelet transform)
of transverse oscillations in a small magnetic element (marked with circles in
Figure 5) at two atmospheric heights sampled by the SuFI/Sunrise 300 nm and Ca
ii H bands (with an average height difference of $\approx 450$ km),
respectively. Islands of high power, particularly those marked by the 95%
confidence level contours, are evident in both wavelet power spectra. The
wavelet co-spectrum and coherence maps of these two power spectra are shown in
the upper- and lower-right panels of Figure 25, respectively. The phase-lag
arrows are over plotted on the entire cross-power spectrum, while the same
arrows are depicted on the latter map only where the coherence exceeds 0.7.
Here, the arrows pointing right represent in-phase oscillations and those
pointing straight up identify 90 degrees phase lags where the oscillations in
300 nm lag behind those observed in the Ca ii H time series. Note here the
changes of phase lags from one time-frequency region to another, particularly,
in regions with confidence levels larger than 95%, and/or areas with coherence
exceeding 0.7 (or 0.8). However, most of the arrows point upwards (with
different angles) in this example, implying an upward wave propagation in the
lower solar atmosphere (i.e., from the low photosphere, sampled by the 300 nm
band, to the heights corresponding to the temperature minimum/low
chromosphere, sampled by the Ca ii H images). A slight downward propagation is
also observed in a small area. These may associate to various wave modes
and/or oppositely propagating waves at different frequencies and times. We
note that such phase changes with time could not be identified using a Fourier
phase lag analysis (see Section 2.2.5), where phase angles are computed as a
function of frequency only.

Whether the cross-power spectrum or coherence should be used for the wave
identification greatly depends on the science and the types of data employed.
While the co-spectrum (which is obtained through multiplying the wavelet power
spectrum of a time series by the complex conjugate of the other) identifies
regions with large power in common (between the two time series), the
coherence (i.e., square of the cross-spectrum normalized by the individual
power spectra; Grinsted et al. 2004) highlights areas where the two time
series co-move, but not necessarily sharing a high common power. An example is
the area around the time and period of 70 and 47 s, respectively, that is
associated to a coherence level exceeding 0.8 (and within the 95% confidence
levels), but with no significant power in the co-spectrum (only one of the
power spectra, i.e., that from the Ca ii H data, show large power at that time
and period location).

As a working example, from the right panels of Figure 25, the phase lag at the
time and period of 75 and 41 s, respectively, reads about 140 degrees, which
is translated to a time lag of $\approx 16$ s. Given the average height
difference of 450 km between the two atmospheric layers, it results in a wave
propagation speed of $\approx 28$ km/s (due to the transverse oscillations in
the small-scale magnetic element). A similar analysis for intensity
oscillations in the same small-scale magnetic element has also been presented
in Jafarzadeh et al. (2017d). Of course, as highlighted in Section 2.2.5,
phase measurements are always subject to an associated uncertainty of
$\pm$360∘ ($\pm 2\pi$), which arises via phase wrapping. As a consequence, to
alleviate ambiguities in phase angles, in addition to subsequently derived
phase velocities, care must be taken to select observational time series where
the atmospheric height separation is not too substantial (see Section 2.2.5
for more discussion), which helps to minimize the ambiguities associated with
phase wrapping.

Depending on science objectives, it may be helpful to inspect the variation of
phase lags with frequency (or period). To this end, a statistical phase
diagram can be created, where all reliable phase angles (e.g., those
associated to power significant at 5%, and/or with a coherence exceeding 0.8)
are plotted as a function of frequency (Jess et al., 2012c). Such a phase
diagram can provide information about the overall wave propagation in, e.g.,
similar magnetic structures. Figure 26 illustrates a phase diagram (i,e, a 2D
histogram of phase angle versus period; from Jafarzadeh et al. 2017d)
constructed from all the reliable phase angles obtained from the transverse
oscillations in 7 small magnetic elements, similar to that discussed above.
The background colors represent the occurrence frequency and the contours mark
regions which are statistically significant (i.e., compared to the extreme
outliers). From this phase diagram, it is evident that the upward propagating
waves (i.e., the positive phase angles in the convention introduced here)
appear preferential.

Figure 27: IMFs $c_{1}\rightarrow c_{8}$, extracted from the original (non-
detrended and non-apodized) HARDcam H$\alpha$ time series overplotted in the
lower-right panel. In addition, the lower-right panel also shows the
polynomial best-fit line (dashed red line) used to detrend the data prior to
FFT/wavelet analyses. It can be seen that the longest period fluctuations
making up IMF $c_{8}$ are similar to the global trend line calculated in
Section 2.2.Note that a summation of IMFs $c_{1}\rightarrow c_{8}$ will return
the original signal.

###  2.5 Empirical Mode Decomposition

Empirical Mode Decomposition (EMD; Huang et al., 1998, 1999) is a statistical
tool developed to decompose an input time series into a set of intrinsic
timescales. Importantly, EMD is a contrasting approach to traditional
FFT/wavelet analyses since it relies on an empirical approach rather than
strict theoretical tools to decompose the input data. Due to the decomposition
being based on the local characteristic timescales of the data, it may be
applied to non-linear and non-stationary processes without the detrending
often applied before the application of Fourier-based techniques (i.e., under
the assumption that such detrending is able to accurately characterize any
non-stationary and/or non-periodic fluctuations in the time series with a low-
order polynomial). As such, it is possible for EMD to overcome some of the
limitations of FFT/wavelet analyses, including aspects of wave energy leakage
across multiple harmonic frequencies (Terradas et al., 2004). non-
stationary/non-period fluctuations that can be characterized by a low-order
polynomial

Figure 28: IMFs $c_{1}\rightarrow c_{8}$ displayed as a two-dimensional map
(left), where yellow and blue colors represent the peaks and troughs,
respectively, of the IMF intensity fluctuations. The horizontal dashed black
lines represent cuts through each IMF, with the corresponding intensity time
series displayed in the right panel. The horizontal dashed red lines represent
the zero value corresponding to each IMF.

Following the methodology described by Terradas et al. (2004), we apply EMD
techniques to the HARDcam H$\alpha$ time series depicted in the upper-left
panel of Figure 6. To begin, extrema in the lightcurve are identified, and are
then connected by a cubic spline fit to provide an upper envelope of the
positive intensity fluctuations (i.e., fluctuations above the mean). Next, the
same process is applied to find the lower envelope corresponding to negative
intensity fluctuations (i.e., fluctuations below the mean). The mean value
between the upper and lower envelopes, at each time step, is denoted
$m_{1}(t)$. The difference between the original input data and the mean
function is called the first component, $h_{1}(t)$. Providing the input time
series contains no undershoots, overshoots, and/or riding waves (Huang et al.,
1998), then the first intrinsic mode function (IMF) is equal to $h_{1}(t)$.

Unfortunately, many input time series contain signal blemishes, and removal of
the first component, $h_{1}(t)$, from the original lightcurve will generate
additional (false) extrema. Hence, to mitigate against these potential issues,
the above procedure is repeated numerous times until the first true IMF is
constructed (see Huang et al., 1998, for more information). The first IMF
constructed, $c_{1}(t)$, is comprised of the most rapid fluctuations of the
signal. This can then be subtracted from the original time series, producing a
residual lightcurve made up of longer duration fluctuations. The process can
subsequently be repeated numerous times to extract additional IMFs until the
amplitude of the residual lightcurve falls below a predetermined value, or
becomes a function from which no more IMFs can be extracted (Terradas et al.,
2004).

Figure 29: Instantaneous frequencies computed from applying a Hilbert-Huang
transform to the HARDcam H$\alpha$ lightcurve shown in the lower-right panel
of Figure 27 and displayed as a function of time (left panel). The solid
purple, blue, dark green, light green, orange, and red lines correspond to
moving average frequencies (computed over a 30 s window) for the IMFs
${c_{2}\rightarrow c_{7}}$, respectively. The vertical error bars correspond
to the standard deviations of frequencies found within the 30 s moving average
windows. The right panel displays the corresponding Hilbert-Huang power
spectrum, calculated by integrating the instantaneous frequency spectra over
time and normalized to the largest power value computed, hence providing a
plot of relative changes in spectral energy as a function of frequency.
Features within the power spectrum are consistent with the FFT and wavelet
outputs shown in Figures 12 & 23.

Figure 27 shows a collection of IMFs extracted from the HARDcam H$\alpha$ time
series depicted in the upper-left panel of Figure 6. It is clear that the most
rapid fluctuations are present in IMF $c_{1}$, with IMF $c_{8}$ documenting
the slowest evolving intensity variations. Plotted on top of IMF $c_{8}$ is
the original H$\alpha$ time series, along with the polynomial best-fit line
(dashed red line) used to detrend the lightcurve in Section 2.2 before the
application of FFT/wavelet techniques. The global trends highlighted by IMF
$c_{8}$ and the polynomial best-fit line are similar, again highlighting the
appropriate use of detrending in Section 2.2, but now compared with
generalized empirical methods. Figure 28 displays the 8 extracted IMFs in the
form of a two-dimensional map, which can often be used to more readily display
the corresponding interplay between the various amplitudes and variability
timescales.

Figure 30: Doppler velocity (left) and circular polarization (CP; center)
$B$-$\omega$ diagrams of a magnetic pore observed in the photospheric Fe i
6173 Å spectral line. The vertical blue dashed lines represents the boundary
of the umbral region as inferred from intensity images. The right panel shows
the average spectra outside and inside the magnetic umbra of the pore. The
five-minute ($p$-mode) oscillations dominate the quiet Sun, but their
amplitude is progressively reduced absorbed as one approaches the concentrated
magnetic fields of the pore, until a series of eigenmodes are excited within
the magnetic tube itself. Image reproduced from Stangalini et al. (2021b).

Once the IMFs have been extracted from the input time series, it is possible
to employ Hilbert spectral analysis (Huang et al., 1998; Oppenheim and
Schafer, 2009) to examine the instantaneous frequencies with time for each
IMF. The combined application of EMD and Hilbert spectral analysis is often
referred to as the Hilbert-Huang transformation (Huang and Wu, 2008). From the
outputs of the Hilbert-Huang transformation, it is possible to display the
instantaneous frequencies for each of the extracted IMFs as a function of
time. The left panel of Figure 29 displays the instantaneous frequencies
corresponding to IMFs $c_{2}\rightarrow c_{7}$ using the purple, blue, dark
green, light green, orange, and red lines, respectively. IMFs $c_{1}$ and
$c_{8}$ have been removed from the plot as these correspond to very high and
low frequency fluctuations, respectively, which clutter the figure if
included. The solid colored lines represent the running mean values of the
instantaneous frequencies (calculated over a 30 s window), while the vertical
colored error bars indicate the standard deviations of the frequency
fluctuations found within the running mean sampling timescale. As already
shown in Figures 27 & 28, the frequencies associated with higher IMFs are
naturally lower as a result of the residual time series containing less rapid
fluctuations. It can be seen the in left panel of Figure 29 that IMF $c_{2}$
contains frequencies in the range of $50-300$ mHz ($3-20$ s), while IMF
$c_{7}$ displays lower frequencies spanning $1-30$ mHz ($33-1000$ s). We must
note that the left panel of Figure 29 is simply a representation of the
instantaneous frequencies present in the time series as a function of time and
does not contain information related to their relative importance (e.g., their
respective amplitudes), although this information is indeed present in the
overall Hilbert-Huang transform.

Finally, it is possible to integrate the instantaneous frequency spectra
(including their relative amplitudes) across time, producing the Hilbert-Huang
power spectrum shown in the right panel of Figure 29. The features of the
Hilbert-Huang power spectrum are very similar to those depicted in the FFT
spectrum shown in the right panel of Figure 12. Notably, there is a pronounced
power enhancement at $\approx 4.7$ mHz, which is consistent with both the FFT
power peaks (right panel of Figure 12) and the heightened wave amplitudes
found at $\approx 210$ s in the wavelet transform shown in the bottom panel of
Figure 23. This shows the consistency between FFT, wavelet, and EMD
approaches, especially when visible wave activity is evident.

###  2.6 Proper Orthogonal Decomposition and Dynamic Mode Decomposition

Figure 31: The left panel shows a sunspot from Jess et al. (2017) in H$\alpha$
intensity using data from HARDcam (one pixel has a width of 0.138”). The
middle panel shows the mean intensity of the time series, the colourbar
displays the magnitude of the mean time series, the solid black line shows
umbra/penumbra boundary (intensity threshold level 0.85) and the green box
(101 $\times$ 101 pixels) shows the region where Albidah et al. (2021) applied
POD and DMD. The right panel displays the PSD of the time coefficients of the
first 20 POD modes (in log scale). The PSD shows peaks between frequencies
$4.3-6.5$ mHz (corresponding to periods of $153-232$ s). Image adapted from
Albidah et al. (2021).

Recently, for the first time, Albidah et al. (2021) and Albidah et al. (2022)
applied the methods of Proper Orthogonal Decomposition (POD; see e.g.,
Pearson, 1901; Lumley, 1967) and Dynamic Mode Decomposition (DMD; see e.g.,
Schmid, 2010) to identify MHD wave modes in a sunspot umbrae. The POD method
defines the eigenfunctions to be orthogonal in space but places no constraints
on their temporal behaviour. On the other hand, DMD puts no constraints on the
spatial structure of the eigenfunctions but defines them to be orthogonal in
time, meaning that each DMD mode has a distinct frequency. Hence, POD modes
are permitted to have broadband frequency spectra but DMD modes are not. This
is shown in the right panel of Figure 31, which shows a broadband power
spectral density (PSD) of 8 POD modes detected in a sunspot umbra by Albidah
et al. (2021) using HARDcam H$\alpha$ intensity observations from Jess et al.
(2017).

Both the POD and DMD produce 2D eigenfunctions as shown in the left and middle
columns of Figure 32, however, they achieve this using different approaches.
Essentially, DMD identifies the spatial modes which best fit a constant
sinusoidal behavior in time, as with a Fourier transform. POD ranks the
spatial modes in order of contribution to the total variance, which DMD cannot
do.

Since POD can produce as many modes as there are time snapshots, the challenge
is to identify which modes are physical and which are not. Similarly, not all
DMD modes may be physical. For practical purposes a physical model, such as
the magnetic cylinder model (see Section 2.9.2 for discussion of MHD wave
modes of a magnetic cylinder) can be used to select POD and DMD modes which
most closely correspond to predicted MHD wave modes. For the approximately
circular sunspot shown in Figure 32, the predicted MHD cylinder modes which
are in the strongest agreement with the selected POD and DMD modes are shown
in the right column. These are the fundamental slow body sausage (top row) and
kink modes (bottom row).

|  |
---|---|---
|  |
Figure 32: The top and bottom rows show snapshots of the slow body sausage and
kink modes, respectively. From left to right, the columns show the POD and DMD
modes from HARDcam H$\alpha$ intensity observations of a sunspot (Jess et al.,
2017), then the corresponding magnetic cylinder model modes. As shown in the
color bar, the intensity oscillations are normalized between $-1$ and $1$,
hence the blue and red regions are in anti-phase. The methods of POD and DMD
provide a most promising approach to decompose MHD wave modes in pores and
sunspots, even if their cross-sectional shapes are much more irregular than
this example. Image adapted from Albidah et al. (2021).

In the case of the magnetic cylinder model, assuming a static background, the
eigenmodes, e.g., kink, sausage and fluting, are orthogonal to each in other
in space by definition. Furthermore, each mode can have a broadband signal in
$\omega$ and $k$ as shown for a real sunspot in the right panel of Figure 31.
Hence, POD can identify such modes in pores and sunspots, providing there is
no significant background flow that will break the condition of orthogonality.
Furthermore, if a mode has a dominant power frequency, this can be identified
with DMD as well as POD. Indeed, this was done by Albidah et al. (2021) for
the 8 POD modes shown in the PSD plot in the right panel of Figure 32 which
have distinct power peaks between 4.3 mHz and 6.5 mHz. In such cases a
combined POD/DMD approach is a most promising avenue for identifying physical
modes. However, it must be highlighted, as initially introduced in Section
2.2, that the characterization of waves using POD and DMD techniques must be
treated with the same caution as traditional FFT approaches. For example, it
is essential that the relative amplitudes of each eigenfunction are compared
to noise and/or background sources to establish its true significance.

As will be shown in Section 3.3, POD and DMD methods are especially useful for
decomposing MHD wave modes in pores and sunspots of more irregular cross-
sectional shapes than the example shown in Figure 32. This is because POD and
DMD do not have the limitation of having their eigenfunctions pre-defined as
they are with Fourier decomposition, where the basis functions are simply
fixed as sinusoids. Even in the standard cylinder model, the eigenfunctions in
the radial direction are Bessel functions not sinusoids. Hence, when it comes
to identifying the spatial structure of individual MHD wave modes in pores and
sunspots, the methods of POD and DMD are more suited to the job than Fourier
decomposition. However, Fourier transforming the time coefficient of a POD
mode is still necessary to calculate its PSD as shown in the right panel of
Figure 31.

###  2.7 $B$-$\omega$ Diagrams

Imaging spectropolarimetry offers the additional possibility to study the
variations in the wave power spectrum as a function of magnetic flux. To this
aim, Stangalini et al. (2021b) have proposed a new visualization technique,
called a $B$-$\omega$ diagram (see Figure 30), which combines the power
spectrum of a particular quantity (e.g., Doppler velocities) with its
corresponding magnetic information. In this diagram, each column represents
the average power spectrum of pixels within a particular magnetic field
strength interval as inferred from polarimetry (e.g., via spectropolarimetric
inversions or center-of-gravity methods; Rees and Semel, 1979). The
$B$-$\omega$ diagram therefore has the capability to help visualize changes in
the oscillatory field as one transitions from quiet Sun pixels outside the
magnetic tube to the inner (more concentrated) magnetic region. In Figure 30
we show an example of $B$-$\omega$ diagram taken from Stangalini et al.
(2021b), which reveals unique wave information for a magnetic pore observed by
IBIS in the photospheric Fe i 6173 Å spectral line. Here, we clearly see that
the the amplitude of five-minute ($\approx$3 mHz) oscillations in the quiet
Sun is progressively reduced as one approaches the boundary of the magnetic
pore (increasing $B$ values). On the other hand, immediately inside the
boundary of the pore (highlighted using a dashed vertical line), a set of
spectral features is observed in both Doppler velocity and CP (circular
polarization) oscillations (i.e., magnetic field oscillations), which are
interpreted as specific eigenmodes of observed magnetic cylinder.

Figure 33: Estimated effects of the spatial resolution (i.e., different FWHMs
of the instrumental PSF; see Equation 12) on the observed Doppler velocity
field. The original Doppler velocity field observed by the CRISP instrument at
the SST in the Fe i 6301.5 Å photospheric spectral line (left panel) is
convolved with a Gaussian PSF with larger and larger FWHMs to mimic the
effects of a lower spatial resolution (middle and right panels). The sign
convention employed shows downflows (positive velocities) as red colors and
upflows (negative velocities) as blue colors. It can be seen in the middle
(${\mathrm{FWHM}}=0.2^{\prime\prime}$) and right
(${\mathrm{FWHM}}=0.5^{\prime\prime}$) panels that progressively worsening
seeing conditions results in lost velocity signals from primarily small-scale
features (e.g., intergranular lanes).

###  2.8 Effects of Spatial Resolution

The solar atmosphere is highly structured, presenting features across a wide
range of spatial scales down to the resolution limit of current
instrumentation. Oscillations can be localized at particular spatial
scales/features (see, e.g., the discussion in Section 3.3). This means that,
for instance, the Doppler velocity, or indeed any other diagnostic, is the
average within the resolution angle of the observations. For this reason, the
signal itself and its inherent temporal oscillations associated with features
below (or close to) the resolution limit can be underestimated (MacBride et
al., 2021).

To illustrate this effect, we consider a case study based on CRISP
observations acquired at the SST of a quiet Sun region, which were previously
deconvolved using the MOMFBD code (van Noort et al., 2005) to reduce the
effects of residual image aberrations. Here, for the seek of simplicity we
consider the starting data as “perfect data” for the only purpose of
illustrating the effects of spatial resolution of the final power spectra of
the oscillations. In the left panel of Figure 33 we show the original
instantaneous Doppler velocity field obtained from the Fe i 6301.5 Å
photospheric spectral line. In order to mimic the effect of a lower spatial
resolution, we convolve this data using a point spread function (PSF), assumed
here to be Gaussian, with a larger full-width at half-maximum (FWHM). In order
to simplify the process, we do not consider the effects of residual seeing
aberrations present in the original and convolved images. Therefore, our PSF
model only considers the effect of the instrumental PSF, which can be
represented by the Houston diffraction limited criterion (Houston, 1927),

${\mathrm{FWHM}}=\frac{1.03\lambda}{D}\ ,$ (12)

where $\lambda$ is the observed wavelength and $D$ is the diameter of the
telescope. Local seeing effects in ground-based observations can further
reduce the effective resolution, in addition to the seeing conditions
themselves varying significantly throughout the observations, thus providing
further (time varying) degradation to the data. In the left panel of Figure
33, the photospheric velocity field is the result of two components: downflows
in the intergranular lanes (red colors) and upflows in the granules (blue
colors). Since the integranular lanes are much smaller and narrower with
respect to the granules, the velocity signals associated with the integranular
regions become more affected (i.e., reduced) by the lower spatial resolution
induced by worsening seeing conditions. This effect is apparent in the middle
and right panels of Figure 33, where the progressively worsening seeing
conditions (${\mathrm{FWHM}}=0.2^{\prime\prime}$ middle panel;
${\mathrm{FWHM}}=0.5^{\prime\prime}$ right panel) result in lost fine-scale
velocity information.

If the resolution angle is smaller than the angular size of the feature being
studied, then the measured signal will approach the true value. This is due to
the ‘filling factor’ being equal to ‘1’, whereby the feature of interest
occupies the entirety of the resolution element on the detector. On the
contrary, if the resolution element is larger than a particular spatial
feature, then the signal measured will be a combination of both the feature of
interest and plasma in its immediate vicinity. Here, the filling factor of the
desired structure is $<1$, resulting in a blended signal that produces the
measured parameters. In the specific case of integranular lanes (see, e.g.,
Figure 33), this means that if the resolution element is larger than their
characteristic width, signal from the neighboring granules will be collected
too. This effect is shown in Figure 34, where the probability density
functions (PDFs) of the instantaneous velocities for different spatial
resolutions is shown. By lowering the spatial resolution, the original skewed
distribution of the velocity, which is a consequence of the different spatial
scales associated with the upflows (blueshifts) and downflows (redshifts), is
transitioned into a more symmetric distribution that is characterized by
smaller velocity amplitudes.

Figure 34: Probability density functions (PDFs) of the instantaneous velocity
fields shown in Figure 33 as a function of spatial resolution. Here, the blue,
orange, and green lines represent the PDFs for three different seeing
conditions represented by a ${\mathrm{FWHM}}=0.16^{\prime\prime}$,
${\mathrm{FWHM}}=0.2^{\prime\prime}$, and
${\mathrm{FWHM}}=0.5^{\prime\prime}$, respectively. It can be seen that worse
seeing conditions (e.g., the green line) produce more symmetric distributions
and smaller velocity amplitudes due to the spatial averaging of the measured
signals.

These effects, in turn, also translate into a reduction of the measured
amplitudes of any oscillations present in the data. This effect can be seen in
Figure 35, where the suppression factor of the Doppler velocity amplitudes
(upper panel) and the resulting power spectral densities in two distinct
frequency bands, namely 3 mHz and 5 mHz (1 mHz bandwidth; lower panel), are
shown as a function of the spatial resolution. The suppression factor gives an
idea of the underestimation of the amplitudes of the embedded oscillations,
and in the top panel of Figure 35 it is normalized to the value associated
with the original SST/CRISP data used here (i.e.,
${\mathrm{FWHM}}=0.16^{\prime\prime}$ provides a suppression factor equal to
1.0). From the upper panel of Figure 35 we can also predict the amplitudes of
the velocity oscillations captured in forthcoming observations from the new 4m
DKIST facility, which could be as large as $1.3-1.4$ times that of the
velocity amplitudes measured with a 1m class telescope at the same wavelength
(under similar local seeing conditions).

Both the suppression factor and the resulting power reduction, as a function
of spatial resolution, are well modeled by an exponential decay of the form,

$A=A_{0}e^{-\frac{{\mathrm{FWHM}}}{{s_{0}}}}+C\ ,$ (13)

where $A_{0}$ is either the amplitude of the velocity signals or the wave
power, $s_{0}$ is a characteristic spatial length, and $C$ is a constant.
Equation 13 characterizes very nicely the impact spatial resolution has on the
visible wave characteristics, whereby when the resolution element is larger
than the characteristic physical scale of the observed process in the solar
atmosphere (i.e., ${\mathrm{FWHM}}>s_{0}$), then the oscillatory signal is
strongly suppressed. This may result in weak oscillatory amplitudes being lost
from the final data products, a process that was recently discussed by Jess et
al. (2021b) in the context of sunspot oscillations.

Figure 35: Wave amplitude suppression factor (upper panel) and the resulting
power spectral densities (lower panel) for observations acquired with
different spatial resolutions. In the upper panel, the wave amplitude
suppression factors (blue dots) are computed with respect to the velocity
information displayed in Figure 33, with the vertical dotted lines
highlighting telescope aperture sizes of 4m (DKIST), 1m (SST), and 0.1m. The
dashed red line displays an exponential fit (using Equation 13), with the fit
parameters shown in the figure legend. The lower panel displays the resulting
power spectral densities, as a function of spatial resolution, for two key
frequencies commonly found in observations of the solar atmosphere, notably
$2.5-3.5$ mHz (orange dots) and $4.5-5.5$ mHz (blue dots). Again, the power
spectral densities are fitted using Equation 13, with the corresponding fit
parameters shown in the figure legend. These panels document the importance of
spatial resolution when attempting to measure weak oscillatory processes,
since poor spatial resolution (either through small telescope aperture sizes
or poor local seeing conditions) may result in complete suppression of the
observable signal.

Such amplitude suppression effects imply that when estimating the energy flux
of waves, one needs to consider the specific spatial resolution achieved and
correct the resulting estimates by a factor depending on the FWHM of the
instrumental PSF and the local seeing effects. We note that this effect
strongly depends on the characteristic spatial length of the processes
observed in the solar atmosphere. In order to illustrate the problem we have
made use of photospheric observations (i.e., Figures 33 – 35). However, due to
the presence of narrow filamentary structures observed in the chromosphere,
the power of the oscillations can be even more underestimated at those
atmospheric heights.

###  2.9 Identification of MHD wave modes

In this Section we will not review MHD wave theory in any great detail since
this has been covered previously in many books and reviews (see e.g.,
Aschwanden, 2004; Nakariakov and Verwichte, 2005; Priest, 2014; Jess et al.,
2015; Roberts, 2019). Instead, we would like to highlight the particular
challenges of identifying MHD wave modes from observational data given what is
known from MHD wave theory.

####  2.9.1 Homogeneous and unbounded plasma

In most textbooks, for simplicity MHD waves are rightly introduced by assuming
a homogeneous unbounded plasma with a straight and constant magnetic field.
This highly idealized plasma configuration only permits propagating Alfvén,
slow, and fast magnetoacoustic wave modes. In stark contrast, the Sun’s
atmosphere is actually very inhomogeneous and the newest high resolution
instrumentation reveal the solar plasma to be ever more finely structured. But
let us assume the wavelengths are large enough so that these MHD wave modes do
not “feel” the effect of any plasma fine structure, hence allowing us to apply
the unbounded homogeneous plasma model, as a zeroth order approximation, to
observational data. How can we actually identify the Alfvén, slow, and fast
magnetoacoustic MHD wave modes? As we shall discuss, in practical terms, even
in this simplest of plasma configurations, each MHD wave mode would actually
be non-trivial to identify without ambiguity, even from excellent quality
spectropolarimetric data.

First, let us consider the Alfvén wave (Alfvén, 1942). The only restoring
force of this wave is magnetic tension, but since this wave is incompressible
the magnetic field lines remain equidistant from each other as they are
oscillating. Hence, although the direction of the magnetic field vectors will
change with time as the field lines oscillate the magnitude of the vectors
will remain constant. Therefore, this wave will not reveal itself through
variations in the magnetic field strength using the Zeeman or Hanle effects.
Also, due to its incompressibility the Alfvén wave would not reveal itself in
intensity oscillations since the density is not perturbed. This only leaves
the velocity perturbations associated with this wave, which could in principle
be detected in Doppler measurements. However, to truly identify an Alfvén wave
it would have to be established that the velocity perturbations were
perpendicular to the magnetic field lines and that the wave vector was not
perpendicular to the direction of the magnetic field. To add even more
difficulty to the challenge of identifying an Alfvén wave, it is only
approximately anisotropic, in the sense that the fastest propagation is along
the direction of the magnetic field and only completely perpendicular
propagation is forbidden, i.e., the more perpendicular the wave vector becomes
relative to the magnetic field the slower the propagation will be.

What about identifying the slow and fast magnetoacoustic modes? The allowed
directions for the slow magnetoacoustic wave vector are very similar to that
of the Alfvén wave, meaning that it is only approximately anisotropic and
propagation perpendicular to the magnetic field direction is forbidden.
However, unlike the Alfvén wave, the slow magnetoacoustic wave is compressible
and should reveal itself in intensity oscillations if the amplitude of the
perturbations are large enough relative to the background. However, to
establish even more convincing evidence, a slow magnetoacoustic wave requires
validation that the plasma and magnetic pressure perturbations are in anti-
phase. Of course, this is not an easy task in observational data and would
require both a fortuitous line-of-sight and an excellent signal-to-noise ratio
to determine perturbations in both intensity and Zeeman/Hanle effect
measurements. In contrast to the Alfvén and slow magnetoacoustic waves, the
fast magnetoacoustic wave is more isotropic in nature since it can also
propagate perpendicular to the magnetic field. A further key difference to the
slow magnetoacoustic wave is that the plasma and magnetic pressure
perturbations associated with a fast magnetoacoustic wave are in phase. To
show this from observational data would provide compelling evidence that a
fast magnetoacoustic wave mode has indeed been identified, but, as with
showing the anti-phase behavior between plasma and magnetic pressures for a
slow magnetoacoustic wave, this is not a trivial task, even with excellent
quality spectropolarimetric data.

Figure 36: Various wave speeds in a flux tube in the lower solar atmosphere,
from the hot ‘NC5’ flux tube model put forward by Bruls and Solanki (1993), in
combination with the surrounding cool VAL-A atmosphere (Vernazza et al.,
1981).

There are also more subtle points in distinguishing between the Alfvén, slow,
and fast magnetoacoustic wave modes depending on the value of plasma-$\beta$,
which itself is difficult to determine from observational data. Importantly,
for MHD wave modes the value of plasma-$\beta$ also indicates the relative
values of the sound and Alfvén speeds. Especially problematic is the case when
the sound speed is close to the Alfvén speed, since here the propagation
speeds of the Alfvén, slow and fast magnetoacoustic waves along the direction
of the magnetic field are practically indistinguishable. This effect is
clearly demonstrated in Figure 36, which is based on the ‘NC5’ flux tube model
presented by Bruls and Solanki (1993), and clearly shows how the localized
velocities associated with different wave modes can become difficult to
disentangle in the lower solar atmosphere, hence providing some ambiguity when
attempting to diagnose the true wave mode from the propagation velocity alone.
But remember, the nuanced discussion we have had here on wave mode
identification assumed that the solar plasma was both homogeneous and
unbounded. In practical terms, it is more likely that the analysis of waves in
the lower solar atmosphere will be directly related to their excitation in,
and propagation through, large scale magnetic structures such as sunspot and
pores (see Section 3.2) or smaller scale structures such as spicules and
fibrils (see Section 3.4). In such cases the most applied model is that of the
magnetic cylinder (e.g., Wentzel, 1979; Wilson, 1979, 1980; Spruit, 1982;
Edwin and Roberts, 1983), which we shall discuss next.

####  2.9.2 Magnetic cylinder model

The advantage of the magnetic cylinder model is that it allows for the key
plasma parameters, e.g., magnetic field strength and plasma density, to differ
inside and outside of the flux tube, allowing us to introduce inhomogeneity in
the direction perpendicular to the cylinder axis. In this model, relative to
the cylindrical coordinates $(r,\theta,z)$, where $r$, $\theta$, and $z$ are
the radial, azimuthal, and axial directions, respectively, waves can either be
standing or propagating in all three orthogonal directions (see the left panel
of Figure 37). If the wave is propagating in the radial direction this is a
so-called “leaky” wave, which is not trapped by the cylindrical waveguide and
damps due to MHD radiation. The so-called “trapped” modes are standing in the
radial direction with the greatest wave energy density in the internal region
of the cylinder. Outside of the cylinder the trapped mode is evanescent and
decays with increasing distance from the tube.

Figure 37: A typical cylindrical flux tube model (left panel) represented by a
straightened magnetic tube of length, $L$, and radius, $R$. The magnetic
field, $B$, is uniform and parallel to the $z$-axis and the whole
configuration is invariant in the azimuthal direction, $\theta$ (labeled as
$\varphi$ in the diagram). In the schematic, the density varies in a non-
uniform transitional layer of width, $l$, from a constant internal value,
$\rho_{i}$, to a constant external value in the local plasma environment,
$\rho_{e}$. The middle and right panels show the effects of $m=0$ (sausage)
and $m=1$ (kink) wave perturbations, respectively, to the equilibrium flux
tube. The sausage wave (middle) is characterized by an axi-symmetric
contraction and expansion of the tube’s cross-section. This produces a
periodic compression/rarefaction of both the plasma and magnetic field. The
kink wave (right) causes a transverse displacement of the flux tube. In
contrast to the sausage wave, the kink wave displacement/velocity field is not
axi-symmetric about the flux tube axis. The red lines show the perturbed flux
tube boundary and thick arrows show the corresponding displacement vectors.
The thin arrows labelled $B$ show the direction of the background magnetic
field. Images reproduced from Arregui et al. (2005, left panel) and Morton et
al. (2012, middle & right panels).

Beyond the basic descriptions of whether the mode is “leaky” or “trapped”, the
azimuthal integer wave number, $m$, defines whether the waves are the so-
called “sausage”, “kink”, or “fluting” modes. The sausage mode has $m=0$ and
is azimuthally symmetric, the kink mode has $m=1$ and is azimuthally
asymmetric (see the middle and right panels of Figure 37). The fluting modes
are higher order in the azimuthal direction with $m\geq 2$. A further
classification of wave types in a magnetic cylinder is “body” or “surface”
modes. A body wave is oscillatory in the radial direction inside the tube and
evanescently decaying outside. Because the body wave is oscillatory inside the
tube, it has a fundamental mode in the radial direction and also higher radial
harmonics. In contrast, a surface wave is evanescent inside and outside of the
tube with its maximum amplitude at the boundary between the internal and
external plasma. Since it is strictly evanescent inside the tube, the surface
mode cannot generate higher radial harmonics.

At this point it will be worth explaining why confusion has arisen over the
years since the seminal publication by Edwin and Roberts (1983), who also
introduced the terms “fast” and “slow” to classify the propagation speeds of
MHD wave modes along the axis of the magnetic cylinder. In the dispersion
diagrams of a magnetic cylinder, distinct bands appear for a particular wave
mode where the axial phase speed is bounded by characteristic background
speeds. As an example, we can model a photospheric waveguide as being less
dense than the surrounding plasma and having a stronger magnetic field
internally than externally. This would be a reasonable basic model for, e.g.,
a pore or sunspot, where the internal density depletion is a result of the
increased magnetic pressure (Maltby et al., 1986; Low, 1992; Cho et al., 2017;
Gilchrist-Millar et al., 2021; Riedl et al., 2021). In this case, we can form
the inequality of the characteristic background speeds as
$v_{A}>c_{e}>c_{0}>v_{Ae}$, where $v_{A}$ is the internal Alfvén speed,
$c_{e}$ is the external sound speed, $c_{0}$ is the internal sound speed, and
$v_{Ae}$ is the external Alfvén speed. This results in a slower band with
phase speeds between $[c_{T},c_{0}]$, where the internal tube speed, $c_{T}$,
is defined as,

$c_{T}=\frac{c_{0}v_{A}}{\sqrt{c_{0}^{2}+v_{A}^{2}}}\ .$ (14)

In addition, a faster band also exists with phase speeds between
$[c_{0},c_{e}]$. Wave modes with phase speeds below the “slow” band and above
the “fast” band are not trapped modes (having real $\omega$ and $k_{z}$
values). The “slow” and “fast” bands for these chosen photospheric conditions
are shown in the dispersion diagram in the left panel of Figure 38.

Figure 38: Left panel: A dispersion diagram is shown for a representative
photospheric magnetic cylinder. It can be seen that there are two distinct
horizontal bands with slower and faster phase speeds. The fast band is bounded
between $[c_{0},c_{e}]$ and the slow band between $[c_{T},c_{0}]$. The
adjectives “slow” and “fast” here have a quite distinct meaning from the terms
slow and fast when referring to the magnetoacoustic wave modes of a
homogeneous and unbounded plasma. Right panel: A cartoon of theoretically
predicted MHD wave modes in a sunspot, and their possible sources, based on
the magnetic cylinder model of Edwin and Roberts (1983). Images adapted from
Edwin and Roberts (1983, left panel) and Evans and Roberts (1990, right
panel).

Although Edwin and Roberts (1983) used the perfectly apt adjectives, “slow”
and “fast”, to describe the phase speed bounds of these distinct bands of
trapped MHD wave modes, they have quite a different physical meaning to the
terms of the slow magnetoacoustic and fast magnetoacoustic waves from the
homogeneous and unbounded plasma model. This is most clearly illustrated when
comparing the same label “fast” in both scenarios. For a cylindrical waveguide
any trapped fast MHD mode is strictly anisotropic since the propagating wave
vector is restricted to being absolutely parallel to magnetic field direction,
which is also aligned with the cylinder axis. However, a fast magnetoacoustic
wave in a homogeneous plasma can propagate with any angle relative to the
magnetic field orientation.

There is a special class of incompressible Alfvén modes that can exist in a
magnetic cylinder with any azimuthal wave number, $m$, the so-called torsional
Alfvén waves (see e.g., Spruit, 1982). Like the Alfvén wave in a homogeneous
plasma, the only restoring force is magnetic tension. However, torsional
Alfvén waves are strictly anisotropic since they can only propagate along the
direction of the tube axis, whereas their counterpart in a homogeneous plasma
can propagate at any angle (with the exception of perpendicular) relative to
the magnetic field. The torsional Alfvén wave can only be excited if the
driver itself is incompressible, meaning that the tube boundary is not
perturbed at all in the radial direction. However, in reality it likely that
the boundary of solar magnetic flux tubes are perturbed to some degree in the
radial direction. If the boundary is only slightly perturbed in the radial
direction, and the dominant perturbations are in the axial direction, then
this will excite a slow mode. If the radial perturbation dominates over the
axial perturbation, resulting in a greater perturbation of the boundary, then
this will excite a fast mode. The greater radial perturbation for a fast mode
means that magnetic tension plays a larger role in the restoring force than
for a slow mode, where the longitudinal compressive forces of plasma and
magnetic pressure dominate.

Understanding the phase relations between the restoring forces for MHD wave
modes in a magnetic cylinder is not as straightforward as it is for the three
possible MHD modes in a homogeneous plasma. This is because the phase
relations between plasma pressure, magnetic pressure, and magnetic tension
restoring forces depend on whether the wave is propagating or standing in each
of the three orthogonal directions, i.e., radial $(r)$, azimuthal $(\theta)$,
and axial $(z)$. Also, the radial spatial structuring of the plasma in a
magnetic cylinder means that perturbed MHD variables, such as the magnetic
field $(B_{r},B_{\theta},B_{z})$ and velocity $(v_{r},v_{\theta},v_{z})$
components, are related, not only by time derivatives, but spatial derivatives
dependent on the variation of the background plasma properties.

A simplified thin tube or “wave on a string” approximation was made by
Fujimura and Tsuneta (2009) to derive the phase relations between $v_{r}$ and
$B_{r}$ for a kink mode, and $v_{z}$ and $B_{z}$ for a sausage mode. This was
done for both propagating and standing waves in the axial direction, but
caution should be taken in applying these results to structures of finite
width. A more detailed investigation into the phase relations of these MHD
variables was done for the sausage mode by Moreels and Van Doorsselaere
(2013), utilizing a magnetic cylinder of finite width under photospheric
conditions. Like Fujimura and Tsuneta (2009), this model predicted the phase
relations for both standing and propagating waves in the axial direction. A
note of caution should be introduced here to state that both the models of
Fujimura and Tsuneta (2009) and Moreels and Van Doorsselaere (2013) assume the
kink and sausage modes are “free” oscillations of the structure and are not
being driven. To correctly derive the phase relations between the MHD wave
variables in a driven system demands that system is solved as an initial value
problem. However, currently the exact spatial and temporal structures of the
underlying drivers of the waves observed in pores and sunspots are not
universally understood.

Although the phase relations between the perturbed variables for any MHD wave
mode may be not simple to predict theoretically, at the least the spatial
structure of these variables (independent of time), providing the cross-
section of the wave guide is resolved (e.g., particularly in the case of
larger magnetic structures such as pores and sunspots), should correlate in
straightforward way. First, let us consider a fixed axial position, $z$, which
for a vertical tube would correspond to a fixed height in the solar
atmosphere. If the magnetic cylinder is oscillating with an eigenmode, then
the variables related to compressible axial motion, i.e., $v_{z}$, $B_{z}$ and
plasma pressure (also related to perturbations in temperature and plasma
density), should have the same spatial structure in the radial $(r)$ and
azimuthal $(\theta)$ directions. Likewise, the spatial structure of variables
related to radial perturbations of the magnetic field, i.e., $v_{r}$ and
$B_{r}$, should be consistent. The same is also true for the variables that
relate to the torsional motions of the magnetic field, i.e., $v_{\theta}$ and
$B_{\theta}$. Again, all these theoretical predictions assume free
oscillations of the entire magnetic structure, e.g., a pore or sunspot. If the
oscillations are being driven, then this is a more complicated and
computationally expensive modeling problem to solve. Also, the spatial scale
of the driver relative to the size of the magnetic structure is crucial. To
excite the global eigenmodes of magnetic structures the driver has to be at
least as large as the structure itself. If the driver is much smaller than the
magnetic structure, it will still excite localized MHD waves, but these will
not be global eigenmodes of the entire magnetic structure. This too requires a
different modeling approach, see e.g., Khomenko and Collados (2006), who
modeled $p$-mode propagation and refraction through sunspots.

Figure 39: Two active regions, NOAA AR12565 (left) and NOAA AR12149 (right),
captured in the G-band by ROSA at the Dunn Solar Telescope. To show the
departure from circular cross-sectional shape, ellipses are fitted to the
sunspot umbrae. The eccentricity of the left umbra is $\epsilon=0.76$, while
the right umbra is $\epsilon=0.58$. Image adapted from Aldhafeeri et al.
(2021). Fast body kink modes, odd $m=1$
---
Fast body fluting modes, odd $m=2$
Fast body fluting modes, odd $m=3$
Figure 40: The normalized density perturbations of fast body wave modes under
representative coronal conditions for the different values of eccentricity
$\epsilon$. Note that the eigenfunctions for slow body wave modes under
photospheric conditions would have a very similar appearance. From top to
bottom, the $m=1$ (kink) and $m=2,3$ (fluting) modes are shown which have an
odd phase structure with respect to the major axis of the ellipse. Image
adapted from Aldhafeeri et al. (2021). Fast body kink modes, even $m=1$
---
Fast body fluting modes, even $m=2$
Fast body fluting modes, even $m=3$
Figure 41: The same wave modes are shown as in Figure 40 but their phase
structure is even with respect to the major axis of the ellipse. Image adapted
from Aldhafeeri et al. (2021).

High resolution images of sunspots, pores, magnetic bright points, and
fibrillar structures are continually telling us that modeling these features
using cylindrical flux tube geometries, while more mathematically simplistic,
is far from realistic. Even from basic membrane models, in which separation of
variables is possible, the cross-sectional shape has a fundamental effect on
the structure of the eigenfunctions. For elliptical magnetic flux tubes,
Aldhafeeri et al. (2021) investigated the effect of eccentricity,
$\epsilon=\sqrt{1-b^{2}/a^{2}}$, where $a$ and $b$ are the semi-major and
semi-minor axes, respectively, on the spatial structure of eigenfunctions.
See, for example Figure 39, which shows two sunspot umbrae fitted with
ellipses with eccentricities $\epsilon=0.58$ and $\epsilon=0.76$. These are
not negligible values since a circle has $\epsilon=0$. Figure 40 shows $m=1$
(kink) and $m=2,3$ (fluting) fast body modes where the phase is odd with
respect to the major axis as eccentricity increases, while Figure 41 shows the
same modes where the phase is even with respect to the major axis. Although
all MHD wave modes in flux tubes of elliptical cross-section have their
spatial structure distorted when compared to their equivalent versions in flux
tubes of circular cross-section, it can be seen that the fluting modes that
have even phase with respect to the major axis (shown in Figure 41) become
notably different in character as eccentricity increases, since previously
distinct regions of phase or anti-phase end up coalescing. This advancement
from the cylindrical flux tube model demonstrates that more sophisticated
modeling of magnetic flux tubes with more realistic, and hence more irregular,
cross-sectional shapes is required to more accurately interpret what type of
wave modes are present in pores and sunspots. Recently this was done by
Albidah et al. (2022) and Stangalini et al. (2022) to identify MHD wave modes
in sunspot umbrae and this will be discussed in Section 3.2.

In Section 2.8 the crucial issue of spatial resolution was discussed. In
smaller scale magnetic structures, such as off-limb spicules or on disc
fibrils, it is not possible to observe the true cross-section of the wave
guide (as is possible for larger on-disc features such as pores and sunspots)
in order to identify eigenmodes. However, fast sausage and kink modes can
still be identified in these smaller structures if the amplitude of the radial
motion (i.e., transverse to the magnetic field direction) is large enough. The
kink mode is the only cylinder mode which causes a transverse displacement of
the axis. For smaller magnetic structures, such as fibrils, the kink mode will
appear as a “swaying” motion. If the radial motion of the fast sausage mode is
large enough, then this causes periodic changes in the width of the structure,
which can be resolved. Wave mode identification in smaller magnetic structures
is addressed in detail in Section 3.4. As for larger scale magnetic
waveguides, where the cross-section can be resolved fully, such as in pores or
sunspots, the right panel of Figure 38 shows the wide variety of theoretically
predicted MHD wave modes, including slow/fast and body/surface, that can exist
in such structures based on the magnetic cylinder model of Edwin and Roberts
(1983). Recent progress in the identification of such wave modes from
observations is discussed in Section 3.3.

Across Section 2, we have discussed the fundamental theoretical considerations
of waves manifesting in the solar atmosphere (Section 2.9), we have provided
an overview of the techniques used to characterize them (Sections 2.2 – 2.7),
and summarized the challenges faced in light of variable spatial resolution
(Section 2.8). Regardless of these challenges, over the last number of decades
the solar community has overcame many obstacles, which has allowed for the
successful acquisition, extraction, and identification of many different types
of wave modes across a wide variety of solar features. In the following
section, we will overview recent discoveries in the field of waves in the
lower solar atmosphere, as well as comment on the difficulties still facing
the global community in the years ahead.

##  3 Recent Studies of Waves

In the past, review articles have traditionally segregated wave activity in
the solar atmosphere into a number of sub-topics based on the specific wave
types and structures demonstrating the observed behavior. For example, Jess et
al. (2015) divided up the review content on a feature-by-feature basis,
including sections related to compressible and incompressible waveforms, which
were subsequently further sub-divided into quiet Sun, magnetic network, and
active region locations. However, as modern observations and modeling
approaches continue to produce data sequences with ever improving spatial
resolutions, placing the physical boundary between two locations becomes even
more challenging. Indeed, emerging (and temporally evolving) magnetic fields
often blur the boundaries between magnetic network elements, pores, proto-
sunspots, and fully developed active regions. Hence, it is clear that solar
complexity continues to increase with each improvement in spatial resolution
made. As a result, dividing the content between previously well-defined
structures becomes inappropriate, which is even more apparent now that mixed
MHD waves (e.g., compressible and incompressible modes; Morton et al., 2012)
are being identified in a broad spectrum of magnetic features.

Hence, for this topical review we employ just three (deliberately imprecise)
sub-section headings, notably related to ‘global wave modes’, as well as
‘large-scale’ and ‘small-scale’ structures. This is to avoid repetition and
confusion, and to allow the overlap between many of the observables in the
Sun’s atmosphere to be discussed in a more transparent manner. Importantly,
while discussing the recent developments surrounding wave activity in the
lower solar atmosphere, we will attempt to pinpoint open questions that
naturally arise from the cited work. We must stress that closing one research
door more often than not opens two (or more) further avenues of investigation.
Therefore, discussion of the challenges posed is not to discredit the cited
work, but instead highlight the difficult research stepping stones facing the
solar physics community over the years and decades to come.

###  3.1 Global Wave Modes

The field of helioseismology has employed long-duration data sequences (some
spanning multiple continuous solar cycles; Liang et al., 2018) to uncover the
internal structure and dynamics of the Sun through its global oscillation
properties. Pioneering observations by Frazier (1968) suggested, for the first
time, the presence of dual oscillating modes in the solar atmosphere,
something that contradicted previous interpretations where the observed
oscillations were simply considered to be an atmospheric response to granular
impacts. It was subsequently shown that a variety of global wavenumbers could
be seen in the photospheric velocity field of the C i $538$ nm line (Deubner,
1975). Importantly, the pioneering work of Deubner (1975) revealed clear
ridges in photospheric $k$-$\omega$ power spectra, which helped to highlight,
for the first time, that the ubiquitous 5-minute $p$-mode oscillations are in-
fact resonant eigenmodes of the Sun. Novel observations acquired during
austral summer at the South Pole discovered global 5-minute global
oscillations at a wide range of horizontal wavelengths, revealing the true
extent of oscillation modes associated with global solar resonances (Grec et
al., 1980; Duvall and Harvey, 1983). Traditionally, in the field of
helioseismolgy, the Sun is considered as an approximate spherically symmetric
body of self-gravitating fluid that is suspended in hydrostatic equilibrium
(Christensen-Dalsgaard et al., 2000). This results in the modes of solar
oscillations being interpreted as resonant vibrations, which can be
represented as the product of a function of radius and a spherical harmonic,
$Y_{l}^{m}(\theta,\phi)$. Here, $l$ relates to the horizontal scale on each
spherical shell (commonly referred to as the ‘angular degree’), while $m$
determines the number of nodes in solar longitude (commonly referred to as the
‘azimuthal order’).

The specific modes of oscillation can be divided up into three main
categories: (1) Pressure modes ($p$-modes), which are essentially acoustic
waves where the dominant restoring force is pressure, providing frequencies in
the range of $\sim 1-5$ mHz and angular degrees spanning $0\leq l\leq 10^{3}$
(Rhodes et al., 1997; Kosovichev, 2011; Korzennik et al., 2013), (2) Internal
gravity modes ($g$-modes), where the restoring force is predominantly buoyancy
(hence linked to the magnitude of local gravitational forces), which typically
manifest in convectively stable regions, such as the radiative interior and/or
the solar atmosphere itself (Siegel and Roth, 2014), and (3) Surface gravity
modes ($f$-modes), which have high angular degrees and are analogous to
surface waves in deep water since they obey a dispersion relation that is
independent of the stratification in the solar atmosphere (Mole et al., 2007).
In the limit that the wavelength is much smaller than the solar radius these
wave are highly incompressible. The main restoring force for $f$-modes is
gravity, which acts to resist wrinkling of the Sun’s surface.

The intricacies of helioseismology become even more complex once isolated
magnetic features, such as sunspots, develop within the solar atmosphere,
which impact the velocities and travel times of the embedded global wave modes
(Braun and Lindsey, 2000; Rajaguru et al., 2001, 2004, 2019; Kosovichev, 2012;
Schunker et al., 2013, 2016). A complete overview of the progress in
helioseismology is beyond the scope of the present review. Instead, we refer
the reader to the vast assortment of review articles that focus on the
widespread development of helioseismology over the last few decades (e.g.,
Deubner, 1983; Bonnet, 1983; Christensen-Dalsgaard, 2002; Gizon and Birch,
2005; Gizon et al., 2010, 2017; Gough, 2013; Basu, 2016; Buldgen et al., 2019)

Importantly, the magnetic field in the solar photosphere is inhomogeneous and
found in discrete concentrations across all spatial scales (Zwaan, 1987).
Outside of the magnetic concentrations, where plasma pressure and gravity are
the dominant restoring forces, longitudinal acoustic waves (i.e., $p$-modes)
are generated at the top of the convection zone from the turbulent motions
constituting the convective motion (Stein, 1967; Goldreich and Kumar, 1990;
Bogdan et al., 1993). The $p$-modes can propagate upwards and contribute to
heating of the higher layers if their frequency is larger than the acoustic
cut-off frequency (Ulmschneider, 1971b; Wang et al., 1995). Thus, the acoustic
waves can dissipate their energy in the solar chromosphere by forming shocks
(as a result of gas-density decreases with height), which are manifested in
intensity images as, e.g., intense brightenings (Rutten and Uitenbroek, 1991;
Carlsson and Stein, 1997; Beck et al., 2008; Eklund et al., 2020, 2021), or
drive, e.g., Type i spicules, in the so-called ‘magnetic portals’ (Jefferies
et al., 2006). Moreover, Skogsrud et al. (2016) showed that these shocks are
associated with dynamic fibrils in an active region they exploited from
observations with SST and the Interface Region Imaging Spectrograph (IRIS; De
Pontieu et al. 2014b) space telescope.

Properties of propagating acoustic/magnetoacoustic waves through the lower
solar atmosphere have also been reported in a number of recent studies from
both ground-based (e.g., Sobotka et al., 2016; Abbasvand et al., 2020a, b) and
space-borne observations (e.g., Martínez-Sykora et al., 2015; Zhao et al.,
2016; Abbasvand et al., 2021). From IRIS observations of a quiet-Sun area in
Mn i 2801.25 Å, Mg ii k 2796.35 Å, and C ii 1334.53 Å spectral lines (sampling
the photosphere, chromosphere, and transition region, respectively), Kayshap
et al. (2018) found upwardly propagating $p$-modes with periods on the order
of $1.6-4.0$ min, and downward propagation in the higher period regime (i.e.,
periods larger than $\approx$4.5 min). Furthermore, Kayshap et al. (2020)
identified the propagation of slow magnetoacoustic waves (with 2-9 min
periodicities), within a plage region, from the high-photosphere/low-
chromosphere to the transition region, using SDO/AIA and IRIS observations.

In addition, $g$-modes (i.e., internal gravity waves) can be produced within
turbulent convective flows (Mihalas and Toomre, 1981, 1982) and propagate
through the lower solar atmosphere, with frequencies shorter than $\approx 2$
mHz (Newington and Cally, 2010; Kneer and Bello González, 2011; Vigeesh et
al., 2017; Jefferies et al., 2019; Vigeesh and Roth, 2020). Their
identifications in the solar atmosphere have, however, been in a challenging
task since they become evanescent in the convection zone and their amplitudes
at the surface are exceedingly small (Schunker et al., 2018; Calchetti et al.,
2021). The internal gravity waves can potentially carry a large amount of
energy flux (of $\approx 5$ kW/m2; Straus et al. 2008) to the chromosphere,
thus contributing to its radiative losses (Vigeesh et al., 2021).

Fortunately, unlike $g$-modes, $f$-modes (i.e., surface gravity modes) have
been detected in abundance and have provided valuable diagnostic information
about flows and magnetic field in the near surface region (Ghosh et al., 1995;
Rosenthal and Christensen-Dalsgaard, 1995; Christensen-Dalsgaard et al.,
1996). Furthermore, $f$-modes have been exploited to quantify the Sun’s
effective seismic (or acoustic) radius (Schou et al., 1997; Antia et al.,
2000; Dziembowski et al., 2001; Dziembowski and Goode, 2005), a relatively new
concept driven by results from helioseismology, as opposed to the measuring
the Sun’s physical (or true) radius. Such studies have shown that $f$-mode
frequencies, as well as being sensitive to the seismic radius, are also
modified by changes in the magnetic field during the solar cycle.

####  3.1.1 Global $p$-modes in the lower solar atmosphere

Acoustic waves (i.e., $p$-modes) can propagate both outside and inside
magnetic concentrations. Through a number of studies prior to the turn of the
century, properties of their ‘global’ oscillations (i.e., properties averaged
over a relatively large field of view) became a “basic fact”, describing the
characteristic periodicity of $p$-modes as 5 minutes in the solar photosphere
(Leighton et al., 1962; Ulrich, 1970; Ruiz Cobo et al., 1997; Schunker et al.,
2009), and 3 minutes in the chromosphere (Evans et al., 1963; Orrall, 1966;
Cram, 1978; Fleck and Schmitz, 1991; Rutten and Uitenbroek, 1991; Carlsson and
Stein, 1992). Standing acoustic waves have also been reported from multi-line
observations in the solar chromosphere (Fleck et al., 1994a), though wave
patterns (and power spectra) were found to be somewhat different in He i 1080
Å observations (Fleck et al., 1994b, 1995), compared to those in other
chromospheric diagnostics (e.g., Ca ii H & K, Ca ii 8542 Å, and H$\alpha$).
While the global $p$-modes are more purely acoustic in nature in the
photosphere, they are more likely to manifest as magnetoacoustic waves in the
upper atmosphere, where the magnetic forces dominate (Khomenko and Calvo
Santamaria, 2013).

Many of the observations demonstrating the characteristic periodicities of the
global $p$-modes have been based on wide-band imaging at low-spatial
resolutions, with very large fields of view. A recent study by Fleck et al.
(2021) highlighted the presence of ubiquitous 3-minute characteristic
periodicities by exploring several advanced state-of-the-art numerical models.
Even so, considerable differences between the various simulations were also
reported, including the height dependence of wave power, in particular for
high-frequency waves, varying by up to two orders of magnitude between the
models (Fleck et al., 2021). Thus, although the numerical simulations provide
us with important information regarding the physical processes embedded within
observational data, they should be interpreted with caution since the
numerical domains are too small to resolve the true physics driving large-
scale global eigenmodes.

Development of modern instruments in recent years, resulting in relatively
narrow-band (often spectrally-resolved) observations at high resolution, have
further explored the highly dynamic nature of the lower solar atmosphere.
These novel observations reveal that the physical properties and structure of
the lower solar atmosphere may significantly vary over different solar regions
(with different levels of magnetic flux and/or topology), as well as through
different atmospheric layers. Therefore, chromospheric wide-band filtergrams,
that often integrate over a significant portion of strong chromospheric lines
(hence, sampling across a large range of heights), may result in mixing (or
averaging) of observable information (e.g., the oscillatory power), which can
largely vary within a short distance in the lower solar atmosphere. A large
variation in the height of formation can also cause a strong temporal
modulation that may consequently destroy the oscillatory signal. Furthermore,
the effect of spatial resolution can be crucial, as information may be lost in
lower resolution observations due to, e.g., smearing (see Section 2.8 for more
discussion related to resolution effects). Moreover, an average power spectrum
over a very large field of view can predominantly be dominated by
characteristics of quiet-Sun regions (which cover the majority of the solar
surface at any given time).

An example of the influence of spatial resolution is the larger (total) energy
flux of acoustic waves (larger by a factor of $\approx 2$) found in a quiet-
Sun region by the 1m Sunrise telescope (Bello González et al., 2010b),
compared to that from the 0.7m VTT telescope (Bello González et al., 2009).
However, the effect of seeing-free observations with Sunrise could also play a
role in that difference, highlighting again the importance of spatial
resolution (as discussed in Section 2.8). Such variations in atmospheric
seeing (that directly affect the spatial resolution achievable) can influence
the measured periodicities, in particular the global $p$-modes that are
ubiquitously visible across the photosphere and chromosphere.

Figure 42: Multi-layer observations of a quiet-Sun region from the low
photosphere to the high chromosphere (left) whose dominant oscillatory periods
are shown on the right. The green, red, and yellow colors in the dominant-
period maps roughly represent periods around 3, 5, and 7 minutes,
respectively. The bottom panels illustrate the corresponding line-of-sight
magnetogram, from Stokes inversions of Fe i 630.2 nm spectral line. Images
reproduced from Samanta et al. (2016).

In the presence of strong magnetic fields (e.g., in network or plage regions,
where a group of concentrated small-scale magnetic features reside), the
global acoustic power is enhanced at photospheric and low chromospheric
heights (known as a ‘power halo’; Brown et al., 1992; Kontogiannis et al.,
2010; Rajaguru et al., 2013), while it is suppressed in the high chromosphere
(so-called ‘magnetic shadows’; Leighton et al., 1962; Title et al., 1992;
McIntosh and Judge, 2001). While the exact mechanisms behind such power
variation are not yet fully understood, a number of suggestions have been
provided in recent years, from both observations and simulations. In
particular, models have shown that the power enhancement at lower heights can
be due to the reflection of fast waves at the magnetic canopy, as a result of
a large Alfvén speed gradient (Khomenko and Cally, 2012; Rijs et al., 2016).
From observations, both magnetic-field strength and inclination have been
found to play an important role, with greater power in the stronger and more
horizontal fields (Schunker and Braun, 2011; Rajaguru et al., 2019). The power
suppression of the acoustic waves in the high chromosphere has been suggested
to be due to the mode conversion at the plasma-$\beta\approx 1$ level (i.e.,
as a result of interactions between $p$-mode oscillations and the embedded
magnetic fields; Moretti et al., 2007; Nutto et al., 2012a), less efficient
wave propagation under the canopy, or the wave-energy dissipation before it
reaches the canopy (Ulmschneider, 1971a; Ulmschneider et al., 2005; Song,
2017; Martínez-Sykora et al., 2020; Srivastava et al., 2021). The power
suppression and its spatial scale has found to be directly correlated with the
magnetic-field strength and/or geometric height (Chitta et al., 2012a; Jain et
al., 2014; Krishna Prasad et al., 2016). From MHD simulations with the Bifrost
code (Gudiksen et al., 2011), Heggland et al. (2011) showed that field
inclination plays an important role in propagation of long-period waves
(longer than 3 minutes) in the solar chromosphere. As such, they primarily
found 3-minute periodicities in regions with weak or vertical magnetic fields
(including the center of strong flux tubes), whereas 5-minute dominant waves
in strong or inclined magnetic fields (such as the edges of flux tubes).

Power suppression of 3 minute oscillations in the upper solar chromosphere has
been reported by Samanta et al. (2016), where almost no oscillatory power at
this period was observed in time series of H$\alpha$ line-core intensity
images from SST/CRISP. The authors, however, found a slightly larger number of
pixels demonstrating 3 minute oscillations in Doppler velocity signatures of
the same spectral line. In addition, they found power halos at lower
atmospheric heights. In this study, the presence of ubiquitous chromospheric
transient events (i.e., short-lived fibrillar structures) was speculated to be
responsible for the power enhancements at lower heights. In addition, it was
speculated that mode conversion was causing the magnetic shadows found around
the 3 minute periodicity in the upper chromosphere. Figure 42 illustrates the
multi-height observations studied by Samanta et al. (2016) (on the left) along
with their corresponding distribution of dominant periods of the oscillations
(on the right), representing periods corresponding to the maximum power at
each pixel. The lack of 3-minute oscillations (i.e., the green color) on the
top layer is evident. Using high-resolution H$\alpha$ line-core observations
with SST, De Pontieu et al. (2007a) had found longer periods in regions where
the field is supposedly more inclined. From spatial distribution of dominant
periods (from a wavelet analysis) they showed that while sunspots and plage
regions were dominated by 3-minute global $p$-modes, 5-minute and longer
periodicities were found in adjacent to the dense plage regions and in more
inclined-field areas, respectively. We should, however, note that such
dominant-period maps demonstrated by Samanta et al. (2016) and De Pontieu et
al. (2007a) should be interpreted with great caution, since multiple peaks
with comparable (or even equal) power may co-exist in a power spectrum. As
such, the period associated to one absolute maximum of the power may not
solely be representative of the oscillations in that pixel. In addition, we
should note that the global wavelet spectrum is often considered a biased
estimation of the true Fourier spectrum, with variable frequency resolution
through the entire spectrum, that can also depend on the choice of wavelet
function (see Section 2.4 for more details).

A recent investigation of such global oscillations (in brightness temperature)
from millimeter observations with ALMA also revealed the lack of 3-minute
oscillations in the solar chromosphere in datasets with relatively large
amounts of magnetic flux (Jafarzadeh et al., 2021). Conversely, the same study
showed the presence of dominating 3-minute oscillations in the most
magnetically quiescent datasets employed. However, due to the uncertain nature
of those millimeter observations, particularly, their exact heights of
formation, further investigations are required. Furthermore, Norton et al.
(2021) reported on global oscillations in the photosphere, from SDO/HMI data,
in various regions, namely, the quiet-Sun, plage, umbra and the polarity
inversion line of an active region. While the 5-minute periodicity, with a
considerably large power, was found in all four areas in Doppler velocity
perturbations, much smaller power enhancements could be observed in intensity
and line-width observations of the quiet and plage regions.

Of particular importance is also the effect of magnetic topology in the
chromosphere, with the multi-layer magnetic canopy whose strength and
thickness depends on, e.g., the magnetic flux involved in their generation
(Jafarzadeh et al., 2017a). By exploring the formation and properties of
various chromospheric diagnostics, Rutten (2017) showed that the dense
canopies of long opaque fibrils in the upper chromosphere, seen in H$\alpha$
line-core intensity images, could act as an ‘umbrella’, obscuring the dynamics
underneath. Thus, this could perhaps explain the lack of 3-minute oscillations
in the high chromosphere (in addition to the magnetic shadows effect). In the
case of ALMA observations, Rutten (2017) speculated that the same phenomena
could also occur, though at those wavelengths the dense fibrillar structures
might not be visible due to their reduced lateral contrast (i.e., an
insensitivity to Doppler shifts; ALMA observes continuum emission, and as such
cannot be used to derive Doppler velocities) We note that similarities between
ALMA observations (at 3 mm) and H$\alpha$ line-width images have been shown by
Molnar et al. (2019).

All in all, it is important to investigate the variation of the global
$p$-modes with height, throughout the lower solar atmosphere, in greater
details. This can hopefully clarify whether the characteristic periodicity
reported in previous studies is constant through the photosphere and the
chromosphere, or whether they vary with height and/or in various solar
regions.

###  3.2 Large-scale Magnetic Structures

Large magnetic structures, in the form of sunspots and solar pores, are
considered ideal laboratories for the study of the excitation and propagation
of MHD waves. Modern high resolution observations have revealed an extremely
complex physical scenario in which different wave modes simultaneously co-
exist in the same magnetic structure, hampering an unambiguous identification
of individual wave modes. This is even more the case for highly structured
magnetic fields, where the wave propagation reflects the geometrical
complexity of the field lines acting as waveguides. However, in recent years
our understanding of MHD waves in large magnetic structures, and their
corresponding role in the heating of the solar atmosphere, has dramatically
changed thanks to the opportunity offered by high-resolution fast-cadence
tomographic imaging and to the new spectropolarimetric diagnostic
capabilities, which have progressively extended up to chromospheric heights
thanks to the technological advances of modern instrumentation. In particular,
the inference of the plasma and magnetic field parameters obtained by
spectropolarimetric inversion techniques have enabled the investigation of the
effects of the magnetic field geometry on the wave propagation itself. In
addition, new spectropolarimetric diagnostics have started to provide
additional information about the magnetic field fluctuations, which are
expected from several MHD wave modes.

The first oscillatory phenomena in the umbra of sunspots were observed by
Beckers and Tallant (1969) and Beckers and Schultz (1972), where the spatially
localized brightenings (so-called ‘umbral flashes’) were immediately
associated with locally excited magnetoacoustic waves propagating upwards
along the field lines. Today, after more than 50 years from the first
discovery of these oscillations, our view of wave excitation and propagation
of MHD waves in large magnetic structures has changed dramatically. From the
observational point of view, in addition to the localized wave phenomena and
disturbances in sunspots and pores (e.g., umbral flashes), the aforementioned
instrumental advances have also allowed the identification of global
eigenmodes of the magnetic structure (e.g., sausage modes, kink modes). These
are generally mixed with other local disturbances, requiring specific
filtering techniques (e.g., $k$-$\omega$ filtering; see Section 2.3) for their
identification. Although most of the literature on the subject mainly reflects
this apparent dichotomy with the two classes of waves (i.e., global and local
oscillations) addressed independently, most recent observations have started
suggesting a superposition of locally excited magnetoacoustic waves resulting,
for example, from $p$-mode absorption or residual local magneto-convection
(Krishna Prasad et al., 2015), and global resonances of the magnetic
structure. These two components can coexist, both of which contribute to the
physical complexity of the observed velocity patterns; an aspect that was
highlighted by Roberts (2019). In the following we wish to strike a balance
between local disturbances and global resonances, and we will summarize the
results from the most relevant studies in recent literature.

Figure 43: Average umbral power spectra for two different sunspots. Solid
lines indicate the power spectra of the chromospheric velocity oscillations
averaged over each entire umbra, demonstrating a peak around 6 mHz. Dashed
lines reveal the photospheric velocity power spectra averaged over each entire
umbra, with a peak around 3.3 mHz and secondary peaks around 6 mHz. Image
reproduced from Centeno et al. (2006b).

####  3.2.1 Magnetoacoustic waves in large-scale magnetic structures

Sunspots and other large magnetic structures, such as solar pores, typically
display intensity and Doppler velocity power spectra that are dominated by
5-minute ($\sim$3 mHz) oscillations in the photosphere, and 3-minute ($\sim$5
mHz) oscillations in the chromosphere (see for instance, Centeno et al.,
2006b, 2009; Felipe et al., 2010; Felipe, 2020; Felipe and Sangeetha, 2020,
and references therein). Of course, it must be noted that the
frequencies/periodicities found at photospheric and chromospheric heights are
not universal values at precisely 3 mHz and 5 mHz, respectively. Indeed,
windows of power are normally referred to when discussing the corresponding
Fourier spectra (Centeno et al., 2006b; Heggland et al., 2011; Gupta et al.,
2013; Khomenko and Collados, 2015), for example, $5\pm 0.5$ minutes ($3.0-3.7$
mHz) and $3\pm 0.5$ minutes ($4.8-6.7$ mHz) for the photosphere and
chromosphere, respectively. These spectral features are depicted in Figure 43,
which clearly shows the frequency transition of peak power between the
photospheric and chromospheric layers of two sunspots. While some authors have
interpreted this to be the combined action of an acoustic cut-off
($\omega_{c}\sim 5.3$ mHz, allowing the upward propagation of magnetoacoustic
waves with $\omega>\omega_{c}$; Deubner and Gough, 1984; Duvall et al., 1991;
Fossat et al., 1992; Vorontsov et al., 1998) and the atmospheric density
stratification resulting in the subsequent amplification of the wave
amplitudes with height, others have explained the spectral features as the
result of the presence of an acoustic resonator (Jess et al., 2021b, 2020;
Felipe et al., 2020). Power spectra similar to those shown in Figure 43 were
also obtained by Kanoh et al. (2016) from observations of a sunspot with
Hinode/SP (in Fe i 6301.5/6302.5 Å) and IRIS (in Si iv 1403 Å), corresponding
to photospheric and transition-region heights, respectively. By comparing
energy fluxes at the two atmospheric regions, Kanoh et al. (2016) speculated
the three orders of magnitude energy decrease with height could suggest wave
dissipation in the chromosphere.

Figure 44: A velocity–time graph extracted from IBIS observations of the
sunspot umbral core on 24th August 2014 by Grant et al. (2018). The horizontal
axis represents Doppler line-of-sight velocity shifts from the rest
wavelength, with the brightnesses displayed correlating to the Ca II 8542Å
spectral profile of a single umbral pixel over the full time series. The left
panel displays velocities of up to 30 km s-1 (or 0.85Å), while the right panel
zooms in to a smaller sub-set for a closer examination of the associated
signatures. The red and green lines in the right panel denote the
accelerations associated with the rising (blue-shifted) and falling (red-
shifted) plasma, respectively. Image reproduced from Grant et al. (2018).

When considering the umbrae of sunspots, the aforementioned umbral flashes
dominated early observations after their initial detection. Umbral flashes
(UFs) initially manifested as intensity brightenings in the core of the Ca ii
K spectral line, subtending multiple arcseconds across an umbra (Beckers and
Tallant, 1969). Consistent with the dominant chromospheric frequencies
mentioned above, UFs exhibited a 3-minute periodicity, however observed
brightness increases of up to 150% (Bogdan, 2000b) and line-of-sight velocity
excursions of 10 km s-1 (Beckers and Schultz, 1972; Phillis, 1975) implied
that these were not the signatures of linear MHD magneto-acoustic
oscillations. Subsequent modelling efforts (see the seminal works of Carlsson
and Stein, 1997; Bard and Carlsson, 2010) established that UFs were the
signature of shocks formed from the steepening of slow magneto-acoustic waves
as they propagate through the large negative density gradients of the low
chromosphere. When these non-linear shock fronts are formed, the intensity
brightenings correspond to the dissipation of wave energy into plasma heat. At
this stage, the plasma is no longer frozen into the magnetic field, and can
propagate isotropically, however as the shocked plasma radiatively cools,
gravitational effects will cause this overdense plasma to infall. The
observational signatures of this morphology can be seen in Figure 44, with
periodic 3-minute intensity brightenings seen in concert with large velocity
excursions consistent with the steepening of slow modes. The right panel of
Figure 44 details the development of the shocked plasma, with the impulsive
shock formation process, characterized by a notable blue-shifted velocity,
leading to a more gradual red-shifted signature due to the infall of the
plasma as it cools. This spectral morophology, known as a ‘saw-tooth’ is
distinctive in comparison to the sinusoidal behavior of linear MHD waves.

The nature of shock development in the solar atmosphere is deserving of its
own dedicated review, as the three characteristic MHD wave speeds lead to a
plethora of potential shock configurations (Delmont and Keppens, 2011). Wave
activity is also not the only driver, with magnetic reconnection capable of
generating a range of shocks (Petschek, 1964; Yamada et al., 2010). The
physical processes involved in shock dynamics entails that current modeling
work still strives to replicate their behavior in realistic conditions (e.g.,
Snow and Hillier, 2019; Snow et al., 2021) and there have only been initial
detections of other modes in the magnetic solar atmosphere (Grant et al.,
2018; Houston et al., 2020). In the context of UFs, it is more instructive to
consider them as a dissipative process of waves, as opposed to wave behavior
synonymous with the focus of this review. It is, however, useful to outline
recent studies that characterize the effect of UFs on umbral plasma and their
effectiveness as wave dissipators. It is also valuable to discuss the effect
UFs have on observables, and their influence on attempts to extract linear MHD
modes from sunspot umbrae.

Figure 45: Plots of stacked Ca ii 8542 Å umbral line spectra grouped by the
neural network classification of MacBride et al. (2021), where the intensity
scale for each spectrum is normalized between ‘0’ and ‘1’ to aid
visualization. A two-dimensional map (lower right) reveals the prominent
neural network classifications for the Ca ii 8542 Å spectra present spatially
across the umbra for a single IBIS spectral imaging scan. Image reproduced
from MacBride et al. (2021).

At the turn of the 21st century, as instrumental capabilities took a leap
forward, initial studies into UF atmospheres still could not resolve any
influence on the umbral magnetic field from shock fronts (Rouppe van der Voort
et al., 2003). Instead, the temperature enhancements of UFs were characterized
by de la Cruz Rodríguez et al. (2013) by applying NICOLE inversion techniques
on Ca ii 8542 Å data, with temperature excursions of up to 1000 K inferred.
However, magnetic field perturbations were still unresolvable, likely due to
the coarse spatial sampling of the data and small sample size of profiles
inverted. The modification of the umbral magnetic field due to UFs was finally
detected by Houston et al. (2018) using polarimetric He i 10830 Å
observations. By sampling this high-chromospheric spectral line with the high
spectral resolution of the FIRS instrument, HAZEL inversions (Asensio Ramos et
al., 2008) revealed $\sim$200 G fluctuations in the vector magnetic field, and
incremental changes in the inclination and azimuth of the field, approximately
8 degrees, implying that the magnetic field enhancement is predominantly along
the direction of wave and shock propagation. Houston et al. (2018) also
corroborated the temperature enhancements of shocks, though with smaller
average values of $\sim$500 K, consistent with the higher atmospheric height
sampled by He i 10830 Å, leading to observation of the shocked plasma as it
enters into its cooling stage. Subsequently, magnetic field changes in Ca ii
8542 Å were reported (Joshi and de la Cruz Rodríguez, 2018), and the
subsequent derivation of semi-empirical UF atmospheric models performed by
Bose et al. (2019).

These results further reinforce the scenario where UFs perturb the magnetic
field geometry of the umbra. However, the perturbation is always predominantly
along the magnetic field vector, as the shock propagates, and the field
returns to its unperturbed state once the shock has propagated through. Thus,
shocks are not seen as candidates to further incline umbral fields to permit
longer period waves to pass, or to greatly impact on adjacent waves as they
propagate. Indeed, the perceived propagation of UF shock fronts horizontally
across the umbra towards the penumbral boundary was instead interpreted as
successive UFs developing along more inclined fields (Madsen et al., 2015),
further implying that the shocked plasma does not play a notable role in
umbral morphology. Despite this, UFs have proved valuable in uncovering fine-
scale umbral features and waves. Henriques et al. (2017) utilized the
brightenings of UFs to reveal small-scale horizontal magnetic fields across
the umbra, revealing a complex ‘corrugated’ structure to the field geometry in
the chromosphere. UFs have been shown to generate a number of plasma flows
with wave implications, notably Henriques et al. (2020) detected downflows,
upflows, and counter-flows before, after, and during the UFs, respectively.
Recently, downflowing UFs have been found to be a signature of standing
oscillations above sunspot umbrae (Felipe et al., 2021).

When considering the processes necessary to balance the chromospheric energy
budget, shocks provide a macroscopic method for converting wave energy
directly into local plasma heating. As discussed earlier, the intensity
excursions of UFs are confirmed as signatures of heating, with between 500 -
1000K temperature increases observed as a result. When UFs are judged in terms
of sole heaters of the chromosphere, Anan et al. (2019) derived a UF shock
heating energy per unit mass of plasma that was insufficient to balance
radiative losses. It is unsurprising that UFs alone are incapable of heating
the chromosphere, particularly as they only occur in localized umbrae.
However, the identification that slow-mode shocks impart heating energy is
notable, given that it is proposed that such wave-driven shocks occur on a
variety of scales across the solar atmosphere (Snow et al., 2021), they
present a viable method to potentially contribute to heating. In addition, as
has been discussed, shock formation is not limited to slow-mode waves. Grant
et al. (2018) observed shocks at the umbra-penumbra boundary of a sunspot that
are inconsistent with the scenario of UF formation, as the inclined magnetic
fields in this region does not produce a large density gradient to steepen
magneto-acoustic waves. Instead, it was proposed that Alfvén waves were
coupling to, and resonantly amplifying, magneto-acoustic waves in at the
penumbral boundary to allow for shock formation, which was verified through
the transverse velocity signatures in the shocks. These Alfvén-induced shocks
produced local temperature enhancements of 5%, less than simultaneous UFs, but
providing dissipation of Alfvén waves in the chromosphere. This further
highlights that the range of possible shock configurations are capable of
dissipating a wide assortment of waves, including the elusive incompressible
Alfvén mode, inferring that there is an tapestry of shock heating across the
atmosphere that is yet to be fully uncovered.

When seeking to observe waves in sunspot umbrae, UFs must always be taken into
account. The intensity excursions associated with UFs have noticable effects
on the spectral profiles sensitive to the density and temperature
perturbations of UFs, such as Ca ii H/K, Ca ii 8542 Å, He i 10830 Å, and upper
chromospheric/transition region channels from IRIS, such as C ii 1335.71 Å, Mg
ii 2796.35 Å, and Si iv 1393.76 Å (Tian et al., 2014; Kayshap et al., 2021).
The line-core emission from these brightenings causes non-trivial profile
shapes to develop, and introduces opacity effects that inhibits velocity
inference through profile fitting (as seen by classes 3 & 4 of Figure 45). A
method of recovering velocity information from these profiles is to use
inversion methods such as NICOLE and HAZEL, however, these are computationally
intensive, time consuming, and do not account for multi-component atmospheres.
The seminal work of Socas-Navarro et al. (2000) showed that in the observation
of a column of shocked plasma, there is always an intermixing of active and
quiescent atmospheres below the resolution limit of the observations. Thus, an
observed spectral profile will in-fact be the result of a two-component
atmosphere, where the filling factor is unknown. It is therefore possible to
extract the linear oscillations embedded in a bright pixel if the atmospheres
can be separated. This was investigated by MacBride et al. (2021) and MacBride
and Jess (2021), who employed machine learning techniques to develop a neural
network capable of classifying Ca ii 8542 Å as a function of line-core
emission (see Figure 45). The authors were then able to separate the two
individual atmospheres through model fitting to provide values for both the
shocked plasma velocity and the associated quiescent component. For any
observer looking into waves in the umbra of a sunspot, care must be taken to
account for UF signatures, either through exclusion of these signatures, two-
component fitting, or the use of spectral lines that are less sensitive to
temperature changes, such as H$\alpha$ (Cauzzi et al., 2008, 2009).

The propagation of waves from the umbrae outward (i.e., along penumbral
filaments) are known as running (penumbral) waves (RPWs; Giovanelli, 1972;
Zirin and Stein, 1972; Brisken and Zirin, 1997; Christopoulou et al., 2000,
2001; Georgakilas et al., 2000; Kobanov and Makarchik, 2004; Bogdan and Judge,
2006; Bloomfield et al., 2007; Sych et al., 2009; Jess et al., 2013; Madsen et
al., 2015; Löhner-Böttcher and Bello González, 2015; Löhner-Böttcher et al.,
2016; Stangalini et al., 2018), which have also been attributed to
magnetoacoustic wave modes (Brisken and Zirin, 1997; Kobanov and Makarchik,
2004). RPW phenomena are mostly prominent in the mid-to-upper chromosphere,
though they have also been observed at photospheric heights (Löhner-Böttcher
and Bello González, 2015; Zhao et al., 2015).

The origin of RPWs has long been debated as either a chromospheric phenomenon
visible as a result of trans-sunspot wave interactions (e.g., Alissandrakis et
al., 1992; Tsiropoula et al., 1996, 2000; Tziotziou et al., 2006; Bogdan and
Judge, 2006; Sharma et al., 2017a; Zhou and Liang, 2017; Priya et al., 2018),
or as the chromospheric signature of upwardly propagating and magnetically
guided $p$-mode waves from the sub-photospheric layers (e.g., Christopoulou et
al., 2000, 2001; Georgakilas et al., 2000; Rouppe van der Voort et al., 2003;
Bloomfield et al., 2007; Reznikova and Shibasaki, 2012; Reznikova et al.,
2012; Jess et al., 2013; Yuan et al., 2014; Madsen et al., 2015). When viewed
as a function of radial distance from the umbral center, RPW signatures
manifest with large apparent phase speeds ($\sim 40$ km/s) and relatively high
frequencies ($\sim 5$ mHz) at the umbra/penumbra boundary, decreasing to lower
apparent phase speeds ($\sim 10$ km/s) and reduced frequencies ($\sim 1$ mHz)
towards the outer penumbral edge (Kobanov et al., 2006). This effect can be
also seen in Figures 46 & 47, which are reproduced from Jess et al. (2013).
Here, both the amplitude and frequency of the captured wave modes is found to
depend strongly on the magnetic field geometry at chromospheric heights. The
dominant frequency of the waves progressively extends towards lower values
(longer periods) as one moves from the center of the umbra and into the
surrounding regions with more heavily inclined magnetic fields (see the
discussions below involving the ramp effect).

Madsen et al. (2015) examined datasets from both the SDO and IRIS spacecrafts
and concluded that the apparent trans-sunspot motion associated with RPWs is
not a real effect, but instead is a result of the waves (originating from the
photospheric $p$-modes) traveling along magnetic field lines of increasing
inclination angle away from the umbral core. On the other hand, Priya et al.
(2018) examined high resolution observations from the Goode Solar Telescope
(GST; Cao et al., 2010). The authors found that oscillatory events in the
sunspot umbra appeared to initiate from earlier occurring RPWs, which, in
turn, caused the development of new RPW events. This was proposed to be
evidence that many of the RPW signatures that are seen at high spatial
resolutions may be entirely chromospheric in origin. However, the authors also
suggest that complex, twisted magnetic field geometry can create a scenario
where wave emergence seems to contradict Madsen et al. (2015). As a result,
with next generation instrumentation and facilities imminent, close attention
will need to be paid to multi-wavelength (i.e., multi-height) observations in
order to compare the small- and large-scale characteristics of RPWs, which
will help to unequivocally determine the underlying physics that underpins
their visible signatures.

Figure 46: Simultaneous images of the blue continuum (photosphere; upper left)
and H$\alpha$ core (chromosphere; upper middle) acquired by the DST using the
ROSA imaging instrument. A white cross marks the center of the sunspot umbra,
while a white dashed line in the continuum image displays the extent of the
photospheric plasma-$\beta=1$ isocontour. The white concentric circles
overlaid on the chromospheric image depict a sample annulus used to extract
wave characteristics as a function of distance from the center of the umbra,
while the solid white line extending into the north quadrant reveals the slice
position used for the time–distance analysis displayed in Figure 47. The
dashed white lines isolate the active region into four distinct regions,
corresponding to the North (N), South (S), East (E), and West (W) quadrants.
The scale is in heliocentric coordinates where $1^{\prime\prime}\approx 725$
km. The remaining panels display a series of chromospheric power maps
extracted through Fourier analysis of the H$\alpha$ time series, indicating
the locations of high oscillatory power (white) with periodicities equal to
$180$, $300$, $420$, and $540$ s. As the period of the wave becomes longer, it
is clear that the location of peak power expands radially away from the center
of the umbra. This effect is synonymous with the presence of running penumbral
waves (RPWs), which were first identified in solar images by Giovanelli (1972)
and Zirin and Stein (1972). Image reproduced from Jess et al. (2013). Figure
47: Top: azimuthally averaged absolute Fourier power displayed as a function
of radial distance from the center of the umbra. Middle: power spectra from
the top panel normalized by the average power for that periodicity within the
entire field of view. Thus, the vertical axis represents a factor of how much
each period displays power above its spatially and temporally averaged
background. Bottom: power spectra normalized to their own respective maxima.
The vertical dashed lines represent the radial extent of the umbral and
penumbral boundaries, while the graduated color spectrum, displayed in the
color bar at the top, assigns display colors to a series of increasing
periodicities between $45$ and $1200$ s. Image reproduced from Jess et al.
(2013). Figure 48: Phase spectrum (upper), coherence (middle; see Section
2.2.2), and amplification (lower) of Doppler velocity oscillations observed in
the photospheric (Si i) and chromospheric (He i) spectral lines in a sunspot
atmosphere. The red line in the upper panel represents the best fit from a
theoretical model. The horizontal dashed black line at a coherence value of
0.7 in the middle panel highlights the lower confidence threshold. The red
line in the lower panel represents the best fit from a theoretical model of
acoustic waves propagating in an isothermal and stratified atmosphere (see
text for more details). Image reproduced from Felipe et al. (2010). Figure 49:
Phase difference spectrum for acoustic oscillations sampled at two geometric
heights. The dashed and dot-dashed lines represent the expected phase as a
function of frequency for non-stratified and vertically stratified isothermal
atmospheres, respectively. The solid line indicates the phase relationship in
a vertically stratified atmosphere including radiative losses from Newton’s
cooling law. For each of the three cases depicted, the same plasma parameters
are utilized ($T=9000$ K, $\Delta{z}=1600$ km, and $g=274$ m s-2). Image
reproduced from Centeno et al. (2006b).

In Figure 48 we show a typical phase diagram from Felipe et al. (2010) that
was obtained by simultaneously measuring the Doppler velocity at both
photospheric and chromospheric heights in a sunspot umbra. Here, we see the
clear effect of an embedded acoustic cut-off, with frequencies above $\sim$5.3
mHz having a positive phase lag, highlighting the upward propagation of these
waveforms. Consequently, these waves experience a rapid density drop as they
propagate into the chromosphere, thus resulting in strong amplification of
their amplitudes (see the lower panel of Figure 48), which eventually results
into shock formation. Following the methodology put forward by Ferraro and
Plumpton (1958) and Centeno et al. (2006b), the amplitude, $A$, of a
monochromatic wave with frequency, $\omega$, in a plane-parallel isothermal
atmosphere permeated by a uniform vertical magnetic field comes from the
solution to the equation,

$c_{s}^{2}\frac{d^{2}A(z)}{dz^{2}}-\gamma g\frac{dA(z)}{dz}+\omega^{2}A(z)=0\
,$ (15)

where $z$ is the vertical coordinate, $g$ is the acceleration due to gravity,
${c_{s}=\gamma gH_{0}}$ is the speed of sound, $H_{0}$ is the pressure scale
height, and $\gamma$ is the ratio of specific heats, which equals $5/3$ for a
monoatomic gas demonstrating adiabatic propagation. A solution to Equation 15
is given by the trial function,

$A(z)=e^{ik_{z}z}\ ,$ (16)

where $k_{z}$ represents the vertical wavenumber. Solving for the vertical
wavenumber, $k_{z}$, provides a dispersion relation of the form,

$k_{z}=\frac{1}{c_{s}}\left(-i\omega_{c}\pm\sqrt{\omega^{2}-\omega^{2}_{c}}\right)\
,$ (17)

where $\omega_{c}=\gamma g/2c_{s}$ is the cut-off frequency. For
$\omega<\omega_{c}$, $k_{z}$ takes imaginary values and the wave is
evanescent. In the opposing regime (i.e., $\omega>\omega_{c}$), waves are able
to propagate. This is illustrated in Figure 49, where the phase angle is
displayed as a function of frequency for waves measured at two independent
geometric heights for both non-stratified and stratified atmospheric models.
The cut-off frequency appears as a natural consequence of the stratification
itself. However, as also shown in Figure 49, a distinct cut-off frequency only
exists in the limit of negligible radiative losses. In more realistic models,
which include aspects of radiative cooling, a sharp separation between the
propagating and evanescent regimes does not exist (solid line in Figure 49),
with the resulting phase diagram displaying a smooth transition at around $3$
mHz. Of course, the above equations that represent the cut-off frequency are
only valid in the limit of an isothermal atmosphere.

The strong vertical stratification of the atmospheric parameters in sunspots
result in orders-of-magnitude changes to the propagation speeds of the
embedded magnetoacoustic waves; namely the Alfvén speed, $v_{A}$, and the
sound speed, $c_{s}$. This, together with the vertical and horizontal
gradients of the background magnetic field, alongside other variations in
local plasma parameters, constitutes important ingredients in the propagation
characteristics of magnetoacoustic waves in these magnetic structures
(MacBride et al., 2022). In Figure 50 we show the variation of the sound and
Alfvén speeds in a typical small sunspot model. To relate these variations to
real observations we must employ approximations, which leads to slightly
different but significant changes (Felipe and Sangeetha, 2020). The cut-off
frequency has been found to significantly change as a function of atmospheric
height, generating important implications for both the heating of the upper
layers of the Sun’s atmosphere and the wave propagation itself (Wiśniewska et
al., 2016; Felipe et al., 2018a).

Further to changes with atmospheric height, it has been shown that the cut-off
frequency depends on the magnetic field inclination, with more inclined fields
allowing the upward propagation of frequencies below the $\sim$5.3 mHz
threshold (Bloomfield et al., 2007; Jess et al., 2013) – the so-called ramp
effect. These waves are generally interpreted as longitudinal slow
magnetoacoustic waves, with the general consensus on their origin being
photospheric, through the absorption of externally driven $p$-modes (see for
instance, Spruit and Bogdan, 1992; Crouch and Cally, 2003; Jess et al., 2012a;
Krishna Prasad et al., 2015), with trans-sunspot oscillations at chromospheric
heights potentially influencing the behaviour of these wave trains (e.g., Chae
et al., 2017; Sych et al., 2020). Although these observational results are in
agreement with the theoretical scenario of propagating slow magnetoacoustic
modes, which in the low plasma-$\beta$ regime correspond to acoustic-like
waves propagating along field lines with the magnetic pressure as the dominant
restoring force, other magnetoacoustic modes exist in spatially uniform
plasmas: namely an incompressible wave with magnetic tension as the sole
restoring force (Alfvén wave), and an intermediate wave mode that can be
thought of as a generalization of an acoustic wave with contributions from
magnetic pressure (fast wave in the presence of a low plasma-$\beta$). In
other words, in low plasma-$\beta$ environments the fast mode is an acoustic
wave modified by the magnetic tension, capable of propagating isotropically
with respect to the magnetic field.

Interestingly, at the equipartition layer where the sound and Alfvén speeds
are nearly equal, a fraction of the energy, $C$, can be either channeled from
a fast mode in the high plasma-$\beta$ regime (which is mainly an acoustic-
like wave) to a fast magnetoacoustic mode in the low plasma-$\beta$ regime, or
converted into a slow mode, thus preserving its acoustic nature. If we
consider the sound speed, $c_{s}=\sqrt{\gamma P_{0}/\rho_{0}}$, and the Alfvén
speed, $v_{A}=B/(4\pi\rho_{0})$, where $\gamma$ is the adiabatic index,
$P_{0}$ is the gas pressure, $\rho_{0}$ is the density, and $B$ is the
magnitude of the magnetic field strength, then the ratio between the two
speeds squared can be given by,

$\frac{c_{s}^{2}}{v_{A}^{2}}=\gamma\frac{4\pi P_{0}}{B^{2}}\ .$ (18)

Substituting the magnetic pressure, $P_{B}=B^{2}/(8\pi)$, into Equation 18 we
obtain,

${\frac{c_{s}^{2}}{v_{A}^{2}}=\frac{\gamma}{2}\beta\ .}$ (19)

This means that the equipartition layer is in practice close to the
plasma-$\beta=1$ surface, and although they are conceptually different, they
are often difficult to segregate from one another in observational data
sequences (e.g., see the discussion points raised by Grant et al., 2018).

Figure 50: Contours of constant density (upper left) and constant magnetic
field strength (lower left) for a typical small sunspot. In the lower-left
panel the labels indicate the magnetic field strength in units of kG, while
the thick black line denotes the isosurface where $v_{A}=c_{s}$. The dotted
lines indicate the geometries of the embedded magnetic field lines. The red
box corresponds to the domain size displayed in the right hand panels, where
contours of constant Alfvén speed ($v_{A}$) and constant sound speed
($c_{s}$), in units of km/s, are depicted in the upper-right and lower-right
panels, respectively. Note the strong horizontal gradients of both $v_{A}$ and
$c_{s}$ due to the Wilson depression. The dotted lines, as per the lower-left
panel, indicate the geometries of the magnetic field lines. Image reproduced
from Khomenko and Collados (2006).

The wave translation process from one form to another is generally referred to
as one of two processes: mode conversion or mode transmission (Crouch and
Cally, 2005; Suzuki and Inutsuka, 2005; Cally and Khomenko, 2015; Pagano and
De Moortel, 2017). Here, ‘mode conversion’ refers to a wave that retains its
original character (i.e., fast-to-fast or slow-to-slow), yet converts its
general nature in the form of acoustic-to-magnetic or magnetic-to-acoustic.
Contrarily, ‘mode transmission’ refers to a wave that preserves its general
nature (i.e., remains a ‘magnetic’ or ‘acoustic’ mode), yet changes character
from fast-to-slow or vice versa. The fraction of energy that can be converted
from fast to slow modes depends on the attack angle of the wave with respect
to the magnetic field lines. The precise transmission coefficient, $T$, is
defined as the proportion of incident wave energy flux transmitted from fast
to slow acoustic waves (Cally, 2001, 2007), which is governed by,

$T=e^{-\pi kh_{s}sin^{2}(\alpha)}\ ,$ (20)

where $k$ is the wavenumber, $h_{s}$ the thickness of the conversion layer,
and $\alpha$ the attack angle itself. The fast-to-fast conversion coefficient,
$C$, can then be obtained by invoking energy conservation: $T+|C|=1$, where
$C$ is a complex energy fraction to take into account possible phase changes
during the process of mode conversion (Hansen and Cally, 2009). We note that
the conversion coefficient, $C$, is larger when the frequency of the incident
waves is higher and the attack angle is larger (Kontogiannis et al., 2014).

Figure 51: A cartoon representation of a sunspot umbral atmosphere
demonstrating a variety of shock phenomena. A side-on perspective of a typical
sunspot atmosphere, showing magnetic field lines (orange cylinders) anchored
into the photospheric umbra (bottom of image) and expanding laterally as a
function of atmospheric height, into the upper atmospheric regions of the
transition region (TR) and corona. The light blue annuli highlight the lower
and upper extents of the mode-conversion region for the atmospheric heights of
interest. The mode-conversion region on the left-hand side shows a schematic
of non-linear Alfvén waves resonantly amplifying magnetoacoustic waves,
increasing the shock formation efficiency in this location. The mode-
conversion region on the right-hand side demonstrates the coupling of upwardly
propagating magnetoacoustic oscillations (the sinusoidal motions) into Alfvén
waves (the elliptical structures), which subsequently develop tangential blue-
and red-shifted plasma during the creation of Alfvén shocks. The central
portion represents the traditional creation of umbral flashes that result from
the steepening of magnetoacoustic waves as they traverse multiple density
scale heights in the lower solar atmosphere. The image is not to scale and is
reproduced from Grant et al. (2018).

Schunker and Cally (2006) have shown how the combination of mode conversion
alongside the ramp effect can result in an acoustic flux which is strongly
dependent on the magnetic field geometry. This was also confirmed by
Stangalini et al. (2011), who found a strong dependence of the wave flux
between the photosphere and chromosphere on the magnetic field geometry
inferred from spectropolarimetric inversions.

Figure 52: Variations of the magnetic field (upper-left), pressure (upper-
right), and velocity (transverse, lower-left; longitudinal, lower-right) at an
elapsed time, ${t=100}$ s, after the beginning of the simulations for a
vertical longitudinal driver with a 10 s (100 mHz) periodicity. The lower 4
panels are identical to the upper 4 panels, only now show the variations for a
horizontal transverse driver with a 10 s (100 mHz) periodicity. In each panel,
the horizontal axis represents the radial distance from the center of the
sunspot, while the inclined black lines highlight the magnetic field
orientation. The two red lines in each panel indicate contours of constant
$c_{s}/v_{A}$, with the thicker red line corresponding to $v_{A}=c_{s}$, and
the thinner red line to $c_{s}/v_{A}=0.1$. The thick black lines inclined
towards the left (with increasing atmospheric height) of each panel indicate
the direction of $\nabla v_{A}$, starting at the location of the pulse. Here,
the $\nabla v_{A}$ line represents the boundary that separates waves
refracting to the right from those refracting to the left, which is
perpendicular to the contours of constant $v_{A}$ at every geometric height.
Image reproduced from Khomenko and Collados (2006).

Grant et al. (2018) have also shown, by exploiting unique high-resolution
observations and magnetic field extrapolations, combined with thermal
inversions and MHD wave theory, that magnetoacoustic waves can couple with
Alfvén waves at the equipartition layer in a sunspot, resulting in Alfvén-
driven shocks that can efficiently contribute to the overall energy budget of
the chromosphere (see Figure 51).

Mode conversion and propagation of magnetoacoustic wave modes in sunspot
atmospheres has also been investigated through numerical two-dimensional
simulations (e.g., Khomenko and Collados, 2008) incorporating realistic
sunspot atmospheres (see Figure 50). In particular, the atmospheric response
to both longitudinal and transverse pulses (with respect to the magnetic field
lines) has been investigated. Figure 52 reveals the velocity, magnetic field,
and pressure fluctuations for a wave with input frequency above the cut-off
value (10 s periodicity or 100 mHz) following 100 s of simulation run time for
both longitudinal and transverse pulses, respectively. It is clear from Figure
52 that the specific input pulse type results in different mixtures of
transverse and longitudinal wave modes, which undergo mode conversion at the
Alfvén/acoustic equipartition layer, and may also be reflected or refracted by
the vertical and horizontal gradients.

From theoretical studies, it was suggested that the enhanced 3-minute wave
power observed at chromospheric heights in sunspot umbrae may come from the
presence of an acoustic resonance cavity, which is established by the
temperature gradients at both the photospheric and transition region
boundaries (see for instance, Hollweg, 1979; Botha et al., 2011; Snow et al.,
2015; Felipe, 2019). Recently, Jess et al. (2020) exploited multi-height high
spatial and temporal resolution observations, spectropolarimetric inversions,
and numerical modeling to provide an observational confirmation of this
physical mechanism. The authors examined the Fourier power spectra originating
within a sunspot umbra and compared this to high-precision simulations
encompassing a variety of different atmospheric stratifications. It was found
that once steep temperature gradients were introduced into the simulation, the
resulting cavity produced resonant amplification of the 3-minute oscillations
(see Figure 53). Following on from the study by Jess et al. (2020), Felipe et
al. (2020) independently confirmed the presence of an acoustic resonator for
another sunspot structure, and highlighted the potential importance of such
findings for future helioseismic investigations of the solar atmosphere.

Figure 53: A vertical stack of narrowband images taken across the Ca ii 8542 Å
spectral line by the IBIS Fabry-Pérot instrument at the DST, revealing the
photospheric (blue) to chromospheric (red) stratification of the sunspot
atmosphere (left panel). This sunspot was found to display characteristics
consistent with the presence of a resonance cavity, which caused the
manipulation of spectral energies across the frequency domain (right panel),
in particular providing a resonant enhancement at $\approx$20 mHz. Image
adapted from Jess et al. (2020).

###  3.3 Eigenmodes of Large-scale Magnetic Structures

MHD theory applied to cylindrical magnetic flux tubes predicts a series of
wave modes (Edwin and Roberts, 1983), which represent the intrinsic overall
response of the magnetic structure to the external forcing. These wave modes
manifest over all scales of magnetic flux tube, however, due to their
irregular cross sectional shapes and corrugated boundaries with surrounding
plasma, sunspots and pores may significantly alter the characteristics of
their resonant modes and their azimuthal properties (Albidah et al., 2021).
Although as in any natural system one may also expect resonant modes in large
scale magnetic structures, such as sunspots and pores, they are still poorly
investigated in these structures. One reason is likely coupled to the fact
that the amplitudes of the oscillations belonging to these modes are
inherently weaker than that of the intensity and velocity excursions
associated with local magnetoacoustic fluctuations such as umbral flashes,
requiring particular filtering techniques for their identification.

Figure 54: Sample images of a near circularly symmetric sunspot, including the
vertical component of the magnetic field ($B_{z}$; upper left), the
photospheric continuum (upper middle), and the chromospheric H$\alpha$ line
core (upper right). The color bar corresponding to the strength of the
magnetic field is saturated at $\pm$1000 G for visual clarity. The lower left
panel displays a snapshot of H$\alpha$ intensities following $k$-$\omega$
filtering (i.e., in both temporal and spatial domains; see Section 2.3.1). The
solid green contour outlines the time-averaged umbra/penumbra boundary, while
the red annulus depicts the extent of the region used for examining azimuthal
wave motion within the umbra, where the center of the annulus is placed at the
center of the umbra. The lower right panel is a time-azimuth diagram following
the polar transformation of the signals contained within the red annulus in
the lower left panel, which allows the circular nature of the wave rotation to
be investigated in a similar way to traditional time-distance diagrams. The
horizontal dashed green line highlights the azimuthal intensity signal
corresponding to the filtered image shown in the lower left panel, while the
solid red line represents the fitted angular frequency (i.e., degrees per
second) of the rotating wave amplitudes. Image reproduced from Jess et al.
(2017).

Figure 55: Left: Phase lag map of the amplitude of circular polarization (CP)
fluctuations at 3 mHz (across a bandwidth of $0.7$ mHz) between the
photosphere and chromosphere. The solid white contour depicts the
umbra/penumbra boundary. The positive phase lags towards the edges of the
sunspot umbra reveal the presence of upwardly propagating magnetic waves.
Right: Phase lag diagram obtained in umbra-penumbra boundary region showing
the presence of a distinct positive peak (downward propagation) in the
spectrum. Plots reproduced from Stangalini et al. (2018).

In this regard, a first attempt to disentangle the signal associated with
resonant modes from the rest of the spatially incoherent oscillations was made
by Jess et al. (2017), who applied three-dimensional Fourier (i.e.,
$k$-$\omega$; see Section 2.3.1) filtering to study the intensity oscillations
of the sunspot (shown in Figure 54) at chromospheric heights. The $k$-$\omega$
diagram of the intensity oscillations displays several horizontal ridges,
which are evident in the right panel of Figure 18. These spectral features are
associated to the spatial scale of the entire sunspot (i.e., at wavelengths
corresponding to the size of the overall sunspot umbra), highlighting the
presence of a spatially coherent oscillations that encompass the entire
sunspot umbra itself. It is worth stressing here that, in order for these
modes to be readily identified, one requires sufficient spectral resolution in
both the temporal and spatial frequency domains. As discussed in Section
2.2.1, to maximize the temporal frequency resolution requires the acquisition
of long duration data sequences. In an analogous manner, large field-of-view
sizes are required to provide sufficient frequency resolution in the spatial
domain (i.e., a small wavenumber resolution, $\Delta{k}$).

After filtering at the wavelengths and frequencies corresponding to the
horizontal spectral ridges shown in the $k$-$\omega$ diagram in the right
panel of Figure 18, Jess et al. (2017) were able to detect a coherent
rotational wave within the sunspot umbra. The observed rotational motion is
visible in the lower panels of Figure 54, where the lower-left panel shows the
reconstructed intensities following $k$-$\omega$ filtering within the solid
black box depicted in the right panel of Figure 18. The lower-right panel of
Figure 54 reveals the time-azimuth map following the polar transformation of
the circular sunspot wave patterns, where the straight diagonal trends
highlight coherent bulk rotations of the MHD wave phenomena. Thanks to
numerical MHD modeling, it was possible to interpret these results as the
first detection of an $m=1$ slow magnetoacoustic mode in the chromospheric
umbra of a sunspot. Building upon the early work of Ulrich (1970) and Deubner
(1975), which has subsequently been improved through the application of more
sensitive instrumentation and modern techniques (e.g., Kosovichev et al.,
1997; Rhodes et al., 1997; Haber et al., 1999; Christensen-Dalsgaard, 2002;
Howe et al., 2004; González Hernández et al., 2006, to name but a few),
significant Fourier power at $\sim 5$ mHz (i.e., consistent with the
generalized $p$-mode spectrum) has demonstrated coherency down to wavenumbers
on the order of $k\sim 0.05$ arcsec-1, corresponding to spatial wavelengths of
approximately $140^{\prime\prime}$ ($\sim 100{\,}000$ km), with radio
observations from missions such as GOLF & BiSON observing global wavenumbers
(e.g., Jiménez-Reyes et al., 2004; Chaplin et al., 2007). This may suggest
that the elevated Fourier power bands shown in the right panel of Figure 18
may be linked to large-scale, sub-surface drivers. While the HARDcam H$\alpha$
observations are chromospheric in nature, being formed at a geometric height
of $\sim 1500$ km (Vernazza et al., 1981; Leenaarts et al., 2012), the highly
magnetic composition of sunspot structures may enable direct and efficient
coupling with the solar layers below (see, e.g., the recent review by Cally et
al., 2016).

Figure 56: Panel a shows a schematic of the spatial structure of the pressure
perturbation for the body mode (left) and the surface mode (right) in both two
and three dimensions. Arrows at the bottom of each schematic show the sausage
oscillation in the flux tube. Panel b shows an example of a body mode in an
elliptical pore. The lower left image shows a G-band intensity image of the
pore, while the image above shows the two-dimensional power plot for the pore
filtered at a central frequency of 11.1 mHz. The white contour here indicates
the location of the pore. The blue cross-cut on the intensity image indicates
the region taken for the one-dimensional power plot across the pore shown on
the right. Panel c shows a similar example for a surface mode. Again, the blue
cross-cut on the intensity image shows the location for the corresponding one-
dimensional power plot shown to the right in this panel. In this instance, the
power plot is produced for the pore filtered at a central frequency of 2.2
mHz, consistent with the timescale of granular evolution. Red dashed lines in
the one-dimensional power plots in both panels b and c indicate the pore
boundary in both instances. The body mode is characterized by a central peak
decaying to the pore boundary, while the surface mode is characterized by
peaks in power at the pore boundary decaying to zero in the center of the
pore. Image adapted from Keys et al. (2018b).

In addition, sausage modes (see Dorotovič et al., 2008; Morton et al., 2011;
Dorotovič et al., 2014; Grant et al., 2015; Freij et al., 2016; Grant et al.,
2022) have also been identified in magnetic pores through variations in their
cross-sectional area of the magnetic structures and associated out-of-phase