Oscillations observed in umbra, plage, quiet-Sun and the polarity inversion line of active region 11158 using Helioseismic Magnetic Imager/Solar Dynamics Observatory data

Using data from the Helioseismic Magnetic Imager, we report on the amplitudes and phase relations of oscillations in quiet-Sun, plage, umbra and the polarity inversion line (PIL) of an active region NOAA#11158. We employ Fourier, wavelet and cross-correlation spectra analysis. Waves with 5 min periods are observed in umbra, PIL and plage with common phase values of ϕ(v, I) = π/2, ϕ(v, Blos) = −(π/2). In addition, ϕ(I, Blos) = π in plage are observed. These phase values are consistent with slow standing or fast standing surface sausage wave modes. The line width variations, and their phase relations with intensity and magnetic oscillations, show different values within the plage and PIL regions, which may offer a way to further differentiate wave mode mechanics. Significant Doppler velocity oscillations are present along the PIL, meaning that plasma motion is perpendicular to the magnetic field lines, a signature of Alvènic waves. A time–distance diagram along a section of the PIL shows Eastward propagating Doppler oscillations converting into magnetic oscillations; the propagation speeds range between 2 and 6 km s−1. Lastly, a 3 min wave is observed in select regions of the umbra in the magnetogram data. This article is part of the Theo Murphy meeting issue ‘High-resolution wave dynamics in the lower solar atmosphere’.


AAN, 0000-0003-2622-7310
Using data from the Helioseismic Magnetic Imager, we report on the amplitudes and phase relations of oscillations in quiet-Sun, plage, umbra and the polarity inversion line (PIL) of an active region NOAA#11158. We employ Fourier, wavelet and crosscorrelation spectra analysis. Waves with 5 min periods are observed in umbra, PIL and plage with common phase values of φ(v, I) = π/2, φ(v, B los ) = −(π/2). In addition, φ(I, B los ) = π in plage are observed. These phase values are consistent with slow standing or fast standing surface sausage wave modes. The line width variations, and their phase relations with intensity and magnetic oscillations, show different values within the plage and PIL regions, which may offer a way to further differentiate wave mode mechanics. Significant Doppler velocity oscillations are present along the PIL, meaning that plasma motion is perpendicular to the magnetic field lines, a signature of Alvènic waves. A time-distance diagram along a section of the PIL shows Eastward propagating Doppler oscillations converting into magnetic oscillations; the propagation speeds range between 2 and 6 km s −1 . Lastly, a 3 min wave is observed in select regions of the umbra in the magnetogram data.

Introduction
The Sun is a seething mass of plasma with a great variety of magnetic fields and electric currents being dynamically generated and distributed throughout its layers. Magneto-hydrodynamics (MHD) waves, generated either by mode conversion of p-modes or excited due to shaking of magnetic flux tubes by turbulent convection and other motions, are thought to contribute to shock heating in the chromospheric layers and, as such, are an important ingredient in heating of the upper atmosphere. Alfvén waves [1] are the least impeded of the MHD waves since they are not reflected by pressure gradients and therefore may reach the corona before dissipating [2] and may play a role in the acceleration of the solar wind.
It is certain that magnetic flux tubes in the solar atmosphere host MHD waves because theory indicates there is no such thing as a pure acoustic wave in a magnetized plasma [3] and also because local helioseismology shows that acoustic shadows exist downstream from sunspots [4,5], meaning that strong magnetic fields decrease the amount of acoustic wave power outgoing horizontally from the region of strong field compared to the acoustic wave power that was observed as incoming. This decrease in acoustic wave power is explained by conversion of acoustic waves into upward or downward propagating MHD waves.
In addition to observing a decrease in acoustic wave power exiting strong magnetic structures, local helioseismology techniques record time-travel changes in waves as they pass through magnetic regions. The changes in the travel times and phase shifts are caused by variations of the sound speed, sub-surface flows and magnetic properties of the medium through which the waves propagate, although it is difficult to differentiate the changes caused by each property [6]. Mode conversion of acoustic waves into MHD waves can introduce phase shifts, too. Numerical modelling indicates that fast magnetic waves are created after acoustic waves are converted at the β = 1 layer [7]. The fast magnetic waves are either reflected downward due to the gradient in the Alfvén speed [8] or converted into upward and downward traveling Alfvén waves [9,10]).
MHD waves contain oscillations in the magnetic field magnitude or direction, so one way to confirm their presence is to observe a time varying component of either. This is not the only way to confirm an MHD wave, though, since area oscillations [11] and spectral line width oscillations [12] have been used in tandem with intensity and velocity oscillations to confirm the presence of MHD waves. Confident detection of magnetic field oscillations is an observational challenge with reports of amplitudes being low, i.e. an upper amplitude of 4 Gauss as observed with the Advanced Stokes Polarimeter in sunspot field strengths for 5 min oscillations [13], 6 Gauss in umbral regions [14], 7-11 Gauss in umbra using an infrared line [15] with only a portion of this amplitude being due to magnetoacoustic waves [16], 4−17 Gauss in pores and network [17] and 20 Gauss in magnetic flux oscillations as estimated by [18] in a study using Hinode SP observations, among other reports in the literature that confirm small amplitudes for δB. Such small wave amplitudes do not imply that the waves carry only small amounts of energy as the energy flux of an MHD wave is proportional to the original field strength, B, times the oscillating components, δB and δv, as described by the Poynting vector.
In addition to searching for oscillations in the magnetic field, phase relations between different quantities are often used to interpret the oscillations as specific wave modes. This was outlined and observed by [19] for magnetic flux and Doppler velocity at different positions on the solar disc and further studied by [20,21]. The observed phases may also indicate that instrumental cross talk [22] or opacity fluctuations sampling the vertical magnetic gradient [15,23] are responsible for the periodic signal instead of MHD waves, although it should be noted that opacity fluctuations are themselves a signature of waves. A more recent development of expected phases for photospheric mode identification was provided for B los  Alfvénic waves. HMI is designed with a high sensitivity to Doppler velocities whereas the line of sight magnetic field measurements of HMI are most likely too noisy to detect the photospheric amplitude of magnetic field changes associated with single mode MHD waves. We should note that the nominal vector magnetic field data from HMI has a cadence of 720 s that precludes detection of the waves in the photosphere that have periods of 5 min or less, but the 135 s vector magnetic field data from HMI has a Nyquist frequency of 3.7 mHz so it can be used for wave research, but we discuss such data options further in §2.
Recently, Alfvén waves heating chromospheric plasma in a sunspot umbra through the formation of shock fronts was reported [37]. In this work, the highly inclined (70-80 degrees) magnetic field geometry in the outer boundary of the umbra, alongside the tangential velocity signatures, distinguished the waves as distinct from umbral flashes. Observed local temperature enhancements of 5% in the chromosphere were reported and thought to be evidence of dissipation of mode-converted Alfvén waves driven by upwardly propagating magneto-acoustic oscillations. Alfvén shocks are predicted to form in regions with high negative Alfvén speed gradients [38]. In this case of the nearly horizontal outer umbral fields, the volume expansion of the magnetic fields and the fact that the density does not drop off steeply means a negative gradient in the Alfvén speed can exist. The study by [37] motivated us to search for oscillatory signatures in highly inclined magnetic field geometries, even though our choice of feature being a polarity inversion line is a very different environment that of the outer umbral field, we have an opportunity to characterize the transverse velocity oscillations.
The purpose of this paper is twofold. First, we explore the oscillations observed with HMI. Since HMI observes the full solar disc for all of the magnetic activity present in Solar Cycle 24, and perhaps that of Solar Cycle 25, then any successful measurement of wave power as well as differentiation of photospheric wave modes from the use of HMI data will support other research conducted with higher spatial and temporal resolution instruments hosted by DKIST, SST and other instruments around the world. Specifically, within this paper, we analyse the signals and oscillations within the active region NOAA 11158 on 16 February 2011 when the region was at South latitude 21 • and West 35 • longitude, a centre-to-limb angle of 40.8 • . We use both the 45-s line-of-sight data and the 135-s vector data [39], a higher-cadence vector dataset whose first release contains about 30 events and 290 h of flaring regions. Second, we observe Doppler velocity oscillations in AR11158 in the region of the polarity inversion line, where the magnetic fields are measured to be transverse to the observer, to detect Alfvénic motions. We are not aware of other MHD wave studies conducted along a polarity inversion line.

Data and methods (a) Data
We use data from the HMI aboard the solar dynamics observatory (SDO). HMI uses the 6173 Å line tuned to six wavelengths to produce 4096 × 4096 full disc images of Doppler velocity, lineof-sight magnetic field, line width and continuum intensity every 45 s with a pixel size of 0.5 [40,41]. This data can be found in JSOC with the names hmi.V_45s, hmi.M_45s, hmi.Lw_45s and hmi.Ic_45s. The continuum intensity is a proxy determined from filtergram sampling away from line centre. The line-widths vary due to thermal width changes and unresolved velocities within the same pixel.
We analysed a 512 × 512 pixel region containing the AR11158 sunspot region on 2011  and indicates the magnetogram flux. M was used to differentiate easily between the vector data quantity of field strength, B, from the inversion. While the line-width values do contain systematic errors [42,43], changes in the thermal width and unresolved velocities within the same pixel increase the line width, and as such, it is valuable if we expect waves such as torsional Alfvén waves, and not compressional waves, to be present. The Stokes I,Q,U,V are also recorded by HMI at the six wavelength positions and by employing a Milne-Eddington inversion code, vector magnetic field maps are produced every 12 min [44,45]. These data can be found with the data name hmi.B_720s with segments that include the field royalsocietypublishing.org/journal/rsta Phil. Trans. R. Soc. A 379:

(b) Neutral line determination
We used a technique that measures the horizontal gradients of photospheric magnetic flux [46,47] in order to identify the polarity inversion line location. First, maps were created with values for regions where the magnitude of the magnetic field was over a certain threshold (20 G). The maps were dilated to allow for overlap of the positive and negative bitmaps, and this overlap was taken to be the polarity inversion line. The locations were stored in an array, the 'PIL bitmap' in which the locations of the neutral line has a value of 1 and elsewhere there was a value of 0. We used the bitmaps, which were determined for every time step, to multiply the Doppler velocity images and other observables in order to create a neutral line data cube of each observable. An example of the neutral line location is shown in figure 2. It is only the long portion between x of 200-300 and y of 280-310 that we consider the stable neutral line for this region. Many of the pixels identified as PIL outside of the long portion are pixels located between plage that are evolving rapidly and as such, are not suitable to study as part of a time-series spanning several hours.

(c) Identification of umbra, polarity inversion line, plage and quiet-Sun subsets
In order to characterize the oscillations in different features within the AR 11158 region, 160 pixels were selected for each of the features of umbra, PIL, plage and quiet-Sun. Data were selected by isolating pixels in the region that matched the following criteria: PIL pixels needed to have field strengths between 600 and 1800 G, inclination between 75 and 105 • , and be located on the PIL in figure    , intensity, I c (counts). Grey-shaded cells are considered noise. Five (three) min periods include 2-4 mHz (5-6 mHz) signals. a Inclinations of plage and small magnetic features, which have been shown to be nearly radial in the photosphere, are larger than expected in HMI data at high centre-to-limb angles, which is due to Stokes Q and U having higher noise than Stokes V and the VFISV inversion weighting the Stokes profiles equally.
of gas to magnetic pressure) for umbra, PIL, plage and quiet-Sun in the mid-photosphere are approximately 0.7, 1-5 (similar to penumbra), 10 2 and 10 4 , respectively [48][49][50][51].  The time series is analysed to determine the cross amplitude and phase angles for the various quantities, including the Doppler velocity and magnetic fluctuations, the Doppler velocity and continuum intensity, etc. To compute the cross spectra, signals are interpolated onto a 10 s grid and shifted past each other in 10 s lag increments up to a ±15.16 min time interval. The resulting cross covariance function is recorded. Restricting the lag interval to ensures wave train coherence time is equivalent to applying a Bartlett window. The Fourier transform of the cross covariance function, the cross spectra, is computed at each position. Phases are determined from the arctangent of the imaginary over the real components of the cross spectra in the filtered frequency range. Taking the absolute value of the magnetic flux before cross-correlating signals ensure similar treatment for field variations, regardless of the magnetic polarity, thus eliminating a 180 •

(e) Wavelet analysis
Wavelet analysis has become a standard tool for identifying periodicities within time series [52]. The benefit of using Wavelet analysis as opposed to traditional Fourier methods is that wavelets allow for the determination of whether the oscillatory power varies over the duration of the observation. In Fourier analysis, the basis functions are localized in the frequency domain, whereas in wavelet analysis, they are localized in both the frequency and time domains. Information is then gathered about the amplitude of periodic signals and how this amplitude varies over the duration of the sampling. For each pixel and observable (v, M or B los , Ic and Lw), we perform a Morlet [52] wavelet transformation and use the 95% significance level to establish that the periods are real. Figure 9 shows representative results from quiet-Sun and umbral pixels while figure 10 shows results from PIL and plage. Thin, vertical white lines are overplotted at the times when the region flared and the cone-of-influence, which denotes edge regions whose results are distorted, is also shown as a thin, white line visible in the lower corners of the plots.

(f) Time-distance slices
In order to observe waves patterns in nearby pixels, we take a 30-pixel slice of data in the x-direction along the direction of the polarity inversion line, across an umbra and in quiet-Sun.
We subtract the average values in each observable so that the oscillation amplitudes are apparent. We then stack these observations in time for 2 h to create time-distance plots as seen in figure 11. The background averages subtracted are shown in figure 12.  table 1 values are reported for the 5 min period for all features, filtering signals to contain the 2-4 mHz frequencies only, prior to calculation. The last row of table 1 investigates the 3 min oscillations seen in approximately 20 umbral pixels, and filters the data for the 5.0-6.0 mHz frequency range prior to calculation. We see suppression of p-mode amplitudes (δV) in the magnetic regions with the Doppler velocity amplitudes decreasing with increasing magnetic field strength such that velocity amplitudes in the quiet-Sun are highest and those in umbra are the lowest. It is suspected that the acoustic waves are being converted into MHD waves here or that the magnetic tension is restricting the flux tubes being pushed around by the external gas motions. In table 1, the I c oscillation amplitudes are similar in that the quiet-Sun shows the most variation, 1.5% and the umbra shows the least, 0.8%. The RMS amplitudes of δB = 35 Gauss and δM = 12 Mx cm −2 (excepting quiet-Sun) are very similar, even though the mean B and M values are quite different between features. It may be that we are sampling the noise floor of the image, or sampling a cross talk, although the power spectra appear to show real, enhanced power in the 5 min range. The line width values are significantly higher in the PIL, with an average of 152 mÅ, than the values in the quiet-Sun, plage or umbral regions, 108, 122 and 128 mÅ, respectively. The higher line width values are indicative of higher thermal widths and unresolved plasma motions. It is worth noting that the line width values are determined from filter positions closer to the core of the spectral line so they represent values somewhat higher in the atmosphere than those determined from the entire line profile, since the continuum is formed lowest in the atmosphere. The δLw values are 1.4-1.8 mÅ excepting quiet-Sun, with the higher value being in the PIL. Remember that RMS amplitudes are 0.7 the peak-to-peak amplitudes.  The average Fourier power computed for the features-the umbra, PIL, plage and quiet-Sunusing 45 s cadence data is shown in figure 6. Significant 5 min power is seen in all features in the velocity data, upper left panel, with amplitudes decreasing with increasing field strengths. There is an artefact at 4 mHz which is the pixel-crossing time and is seen as a peak in magnetogram power in umbra, plage and PIL, in intensity power in the umbra and PIL data and in the umbral Lw data due to spatial gradients in the magnetic, intensity and line width data. The tracking algorithm, mtrack, was employed to return a 0.03 • pixel size and at −21 • latitude, a pixel of 0.03 • subtends 364 km. At this latitude, the expected differential rotation rate is approximately 1.57 km s −1 but the sunspot rotates slightly slower at 1.46 km s −1 and as such, it takes roughly 200 s for any feature to rotate into the next pixel, hence the 4 mHz power. The quiet-Sun magnetogram power (red line, upper right panel) represents pure noise. There is a peak at 5.6 mHz in the umbral magnetogram power that represents the 3 min mode. The intensity power spectra (lower left panel in figure 6) shows 5 min wave power in the quiet-Sun, some reduced 5 min power in the plage, and the 4 mHz pixel crossing artefact in the PIL and umbra data. The line width power spectra (lower right panel) shows a bump of enhanced power in the 5 min band   for quiet-Sun and umbrae with perhaps some 3 min wave power in the umbra, too, but it is fairly noisy. There is a slight enhancement of 5 min power for the plage and PIL data, too. The average Fourier power computed for the features-the umbra, PIL, plage and quiet-Sunusing 135-s cadence data from the vector field inversion is shown in figure 7. Significant 5 min power is seen in all features in the velocity data, upper left panel, with amplitudes decreasing with increasing field strength features. Five minute power is seen in the umbral field strength and Doppler width power spectra but very little other signal is seen. The azimuthal power spectra show no excess power, similar to the inclination power (not shown). Phases are not calculated from the 135 s data as it is deemed too noisy for the phase values to be coherent.

Results and discussion
The mode, or most frequent value, of the phase for each feature and observable pair is shown in table 2. For all features, φ(v,I) = 110-130 • , indicating that the velocity signal leads the intensity signal for somewhat greater than a quarter π/2 of the approximately 5 min period, i.e. a lead of 120 • for a 300 s period is 100 s. By contrast, the most frequent phase values of φ(v, |M|) = −90-110 • for PIL, umbra and plage indicate that the velocity signal lags the absolute value of the magnetic signal by a bit more than π 2 . To interpret the phase values as wave modes, we use table 1 from [24] whose mathematical framework predicts that a φ(B los , v) = ±π/2 with a φ(v,I) = ±π/2 and φ(I,B los ) = 0, π are signatures of sausage waves that are slow standing or fast standing surface modes. The observed quantities, as used for determining these expected phase values, are lineof-sight velocity and magnetic flux. The HMI phase values shown in corresponding columns of table 2 are consistent with these wave modes.
Phases between the line-width and other observables vary more between the different features. φ(I,Lw) of umbra and PIL peak between 70-110 • while plage and quiet-Sun peak at −130 • , see figure 8 lower left panel and corresponding column in table 2. This indicates mechanisms that are 180 • out of phase from each other in intensity and line width variations. φ(Lw,|M|) values are 110 • for PIL and 30 • for plage, indicating modes that are π/2 out of phase from each other. This could be geometric because the PIL is horizontal to the observer's line of sight with wave motions being perpendicular to the field direction whereas the plage is more radial and we are observing changes along the flux tube in the z-direction.

(b) Wavelet analysis
Sample wavelet plots are shown in figures 9 and 10. In figure 9, a quiet-Sun signal is shown in the top four panels and compared with an umbral signal shown in the bottom four panels. The quiet-Sun velocity shows 5 min power peaking around 3.3 mHz, but ranging from 2.5 to 4.5 mHz, and the umbral panel shows significant power in a wider frequency band, from 2.5 to 5.8 mHz. Very little low-frequency power is seen in the umbral velocity. Less power is seen in the umbral intensity data than the quiet-Sun intensity, which is expected as the p-modes do modulate the intensity in the quiet-Sun. For the umbra, the magnetic wavelet panels show power in the 3 min band (between 5 and 6 mHz) which is unexpected since the 3 min umbral oscillation has only       The period of oscillations in magnetic flux across the PIL (left column, middle row panel in figure 11) appear larger than those in the Doppler oscillations (left column, top row panel). If the magnetic oscillations, which are likely a signature of Alfvén waves that have converted from acoustic oscillations along the right side of the PIL, then the differing periods of these two oscillation signals may contain important information on the nature of this conversion process. A relevant theoretical aspect to consider in this regard may be the resonant conversion of acoustic to Alfvénic waves at photospheric heights which has been discussed by [53,54] who both report findings that a sound wave is coupled to an Alfvén wave with double period and wavelength when the sound and Alfvén speeds are equal. Further investigation into the periods of magnetic and Doppler oscillations along the PIL are warranted.
In the middle column of figure 11, the 3 min umbral oscillations, with low amplitudes, can be easily seen in the Doppler data (top row, middle panel) and the 5 min oscillations can be seen with high amplitudes in the quiet-Sun Doppler data (top row, right panel). Very little line width oscillations are seen in the umbra but strong line widths oscillations are seen in the quiet-Sun. The PIL line width variations have finer structure as compared to the quiet-Sun line width variations, see lower row. The umbral and quiet-Sun time distance data show 3-and 5-min oscillations but no similar behaviour of acoustic to magnetic conversion, so we assume the geometry of the PIL with the horizontal, confined field lines provides an unusual environment for wave propagation.
Of course, a concern in reporting amplitudes, power and phase values from a filtergraph type of instrument is that the time-varying signal is not measured per se, but is due to opacity or adiabatic fluctuations which then alter the height of formation or line profile characteristics in a way that cause periodic signals in the observables. Simulations need to be carried out with the 6173 Å Fe-I line and the HMI data processing algorithms to quantify these effects, but this effort is outside the scope of this paper.

Conclusion
HMI provides a plentitude of data and the potential for its use is immense. In particular, the relatively high cadence of HMI for a full-disc imager allows one to see temporal variations in a large number of observables and thus makes it possible to study waves at frequencies relevant for the p-mode band and the 3 min oscillations that permeate the photosphere and chromosphere and are suspected of driving at least some of coronal dynamics and energetics. This paper represents only a small fraction of what can be done with the data to investigate the presence of MHD waves in the photosphere.
In future efforts, we hope to conduct an analysis similar to that in this paper for an active region near the disc centre, or as an active region crosses the disc, to understand how the centre-to-limb position affects the results. Also, we hope to repeat this analysis using data corrected for scattered light.
In the vicinity surrounding AR 11158, and in the active region itself, we find the expected 5 min power evident in the velocity and intensity data for quiet-Sun, see figure 6. The same is true for plage, although the power in the intensity is not as strong and we begin to see enhanced power in the magnetogram for plage. We also find significant oscillations in the 5 min band for velocity signals from umbral and PIL pixels. It is less evident, but power is enhanced in the 5 min band for plage, PIL and umbral pixels in the magnetogram data and for all features in the line-width data. The 135 s vector data confirms the umbral 5 min oscillations in magnetic field and line width as the Doppler width and field strength of the vector data show a significant 5 min peak, see figure 7.
Surprisingly, for some umbral locations, a peak is seen in the magnetogram signal at the 3 min period, around 5.6 mHz, in both the Fourier power (figure 6) and wavelet power spectra (figure 9).Observation of 3 min Doppler power has been reported in the photosphere in numerous studies, including IBIS observations of a pore in the Fe I 617.3 nm line showed that 3 min wave was already present at the photospheric height of formation of this line [55]. The umbral 3 min waves observed in the chromosphere were shown to be already present in the photosphere in a study by [28]. Numerical simulations [56] aided in the conclusion that propagation of waves in the 3 min band directly from the photosphere can explain the observed chromospheric 3 min oscillations. So it is not shocking to see the 3 min power, the surprising part is that it appears in the magnetogram data.
Phase values of HMI oscillations in φ(v,I), φ(v, |M|) and φ(I, |M|), shown in table 2, observed in umbra, plage and PIL are in agreement with a mathematical framework of these waves being slow standing sausage modes. This finding is consistent with numerous other studies which have interpreted 5 min oscillations in magnetic structures in the photosphere as slow standing modes.
A time-distance diagram for a section across the PIL shows Doppler oscillations progressing Eastward at ∼2.7 km s −1 , with magnetic oscillation amplitudes increasing as the Doppler amplitudes damp, see figure 11, left column, top and middle rows. The magnetic disturbances then propagate at 2-6 km s −1 . Enhanced line widths are found at the locations where the waves change from being primarily acoustic to primarily magnetic. The umbral and quiet-Sun time distance data show 3 and 5 min oscillations but no similar behaviour of acoustic to magnetic conversion, so we assume the geometry of the PIL with the horizontal, confined field lines provides an unusual environment for wave propagation and a clear indication of the possible existence of Alfvén modes being generated and propagating in the PIL photosphere.
While the amplitudes of oscillations and phase relations in HMI data reported herein support the presence of MHD waves in and around the active regions, forward modelling of the spectral line dynamics in the presence of realistic MHD modes should be carried out prior to a final confirmation that the values reported in this paper are solar and not instrumental artefacts or crosstalk.
Future collaboration within the waves in the lower solar atmosphere (WaLSA) group will help to isolate and characterize the photospheric wave properties here shown to be abundantly contained within the HMI data in order to support higher resolution studies from other instruments. Funding. This work was supported by NASA contract NAS5-02139 to Stanford University (HMI, PI P.H. Scherrer) and by a NASA Guest Investigator contract 80NSSC18L0668 (MHD Waves, PI J. Zhao).