THE SOURCE OF 3 MINUTE MAGNETOACOUSTIC OSCILLATIONS IN CORONAL FANS

The observational data presented here are part of a sequence obtained during 13:32–14:04 UT on 2011 July 13, with the Dunn Solar Telescope (DST) at Sacramento Peak, New Mexico. We employed the Rapid Oscillations in the Solar Atmosphere (ROSA; Jess et al. 2010a) camera system to image a 69'' × 69'' region encompassing active region NOAA 11250, positioned at heliocentric coordinates (−146'', −486''), or S26.3E10.1 in the conventional north–south–east–west coordinate system. A spatial sampling of 0069 per pixel was used for the ROSA cameras, to match the telescope's diffraction-limited resolution in the blue continuum to that of the CCD. This results in images obtained at longer wavelengths being slightly oversampled. However, this was deemed desirable to keep the dimensions of the field of view the same for all ROSA cameras.

A recent addition to the DST's imaging capabilities is a new high quantum efficiency device, the Hydrogen-Alpha Rapid Dynamics camera (HARDcam). This camera, an iXon X3 DU-887-BV⁵ model manufactured by Andor Technology, consists of a back-illuminated, 512 × 512 pixel² electron-multiplying CCD, with a quantum efficiency exceeding 95% at 6500 Å. As a result, this camera is best suited to imaging in the red portion of the optical spectrum. The triggering and readout architectures are identical to the first-generation ROSA instrumentation, allowing HARDcam to be seamlessly integrated with the existing setup. To optimize the exploitation of its high quantum efficiency, HARDcam was employed behind a 0.25 Å Hα core filter, incorporating a spatial sampling of 0138 per pixel, providing a field-of-view size (71'' × 71'') comparable to existing ROSA image sequences.

In addition to ROSA and HARDcam observations, the Interferometric BIdimensional Spectrometer (IBIS; Cavallini 2006) was used to simultaneously sample the Ca ii absorption profile at 8542.12 Å. IBIS employed a spatial sampling of 0097 per pixel, which allowed ROSA's near square field of view to be contained within the circular aperture provided by IBIS. A total of 13 discreet wavelength steps, with 10 exposures per step to assist image reconstruction, were used to provide a complete scan cadence of 43.4 s. A white-light camera, synchronized with the IBIS feed, was utilized to assist the processing of narrowband images. Full details of the observations presented here, including filters and exposure times used, can be found in Table 1, while sample images can be viewed in Figure 1.

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During the observations, high-order adaptive optics (Rimmele 2004) were used to correct wave-front deformations in real time. The acquired images were further improved through speckle reconstruction algorithms (Wöger et al. 2008), utilizing 32 → 1 restorations for the G-band and 4170 Å continuum images. The remaining HARDcam and IBIS images were processed with 35 → 1 and 10 → 1 restorations, respectively. Post-reconstruction cadences are displayed in the fourth column of Table 1. A full image-reconstructed IBIS scan through the Ca ii absorption line includes a blueshift correction, required due to the use of classical etalon mountings (Cauzzi et al. 2008). Atmospheric seeing conditions remained excellent throughout the time series. However, to ensure accurate co-alignment in all bandpasses, broadband time series were Fourier co-registered and de-stretched using a 40 × 40 grid, equating to a ≈17 separation between spatial samples (Jess et al. 2007a, 2008a). Narrowband images, including those from IBIS, were corrected by applying de-stretching vectors established from simultaneous broadband reference images (Reardon & Cavallini 2008; Jess et al. 2010b; Reardon et al. 2011).

The Atmospheric Imaging Assembly (AIA; Lemen et al. 2011) onboard the Solar Dynamics Observatory (SDO; Pesnell et al. 2011) was utilized to provide simultaneous EUV images of active region NOAA 11250. This instrument images the entire solar disk in 10 different channels, incorporating a two-pixel spatial resolution of 12, and a cadence of 12 s. Here, we selected seven EUV data sets spanning 13:30–14:05 UT on 2011 July 13, consisting of 175 images in each of the 94 Å, 131 Å, 171 Å, 193 Å, 211 Å, 304 Å, and 335 Å channels. In addition, one contextual 4500 Å continuum image, acquired at 14:00 UT, was obtained for the purposes of co-aligning AIA data sets with images of the lower solar atmosphere.

The EUV bandpasses were specifically chosen to cover a wide range of transition region and coronal temperatures, spanning approximately 50,000 K–7 MK, under non-flaring conditions. Transition region imaging is covered by the He ii-dominated 304 Å bandpass, with a typical formation temperature of ∼50, 000 K (Jess et al. 2008c). The selected coronal channels, consisting of the 94 Å, 131 Å, 171 Å, 193 Å, 211 Å, and 335 Å bandpasses, demonstrate typical effective temperatures of approximately 7.0 MK, 0.4 MK, 0.7 MK, 1.6 MK, 2.0 MK, and 2.8 MK, respectively (O'Dwyer et al. 2010; Brooks et al. 2011). Thus, the AIA data sets, in conjunction with our ground-based observations of the lower solar atmosphere, provide us with the ideal opportunity to investigate the coupling of lower atmospheric (photosphere and chromosphere) phenomena with their multi-million degree coronal counterparts, at unprecedented spatial and temporal resolutions.

The AIA data were processed using the standard aia_prep routine, and include the removal of energetic particle hits, in addition to the co-registration of images from different wavelengths on to a common plate scale. Subsequently, 200'' × 200'' sub-fields were extracted from the processed data, with a central pointing close to that of the ground-based image sequences. A sub-field image, including an outline of the field of view obtained with ground-based observations, is shown in the upper-left panel of Figure 2. Using the AIA 4500 Å context image to define absolute solar coordinates, our ground-based observations were subjected to cross-correlation techniques to provide sub-pixel co-alignment accuracy between the imaging sequences. To do this, the plate scales of our ground-based observations were first degraded to match that of the AIA image.⁶ Next, squared mean absolute deviations were calculated between the data sets, with the ground-based images subsequently shifted to best align with the AIA reference image (see the lower panels of Figure 2 for interlaced examples). Following co-alignment, the maximum x- and y-displacements are both less than one-tenth of an AIA pixel, or 006 (≈45 km).

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To investigate oscillatory phenomena occurring in the sunspot umbra, we first masked out all other areas of the field of view. This was deemed desirable so that intrinsically faint umbral structures would become much more apparent in scaled intensity images. To isolate the umbra, we first had to create a binary map, whereby umbral pixels were given a value of "1", and non-umbral pixels assigned a value of "0." This form of binary map was created by first averaging the 4170 Å continuum images over the entire 32 minute duration of the data set. Next, the umbral pixels were defined as those with an intensity below 45% of the median granulation intensity, and subsequently assigned a value of "1." This accurately defined the perimeter of the dark umbra. However, some small-scale structures, which existed inside the umbra (e.g., umbral dots (UDs)), exhibited higher relative intensity values. Therefore, to include all structures which existed inside the umbral boundary, all pixels which lay inside the outer perimeter were assigned a value of "1," while all other pixels were given a value of "0." The time-averaged 4170 Å continuum image, including the outline of the umbral binary map, is shown in the left panel of Figure 3.

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Examination of the time-averaged umbral intensity map (middle panel of Figure 3) revealed a collection of near-circular brightenings. These brightenings are consistent with the signatures of UDs, whereby they typically exhibit diameters between 02 and 08, and intensities 1.3–1.5 times brighter than the background umbral value (Denker 1998; Tritschler & Schmidt 2002; Sobotka & Hanslmeier 2005). Due to their sharp appearance in a time-averaged intensity image, it hints at the persistent nature of such structures in the same spatial location, either through long lifetimes (e.g., 30 minutes or more; Sobotka et al. 1997a, 1997b), or by the reappearance of new UDs in the same spatial location. Analyzing simultaneous G-band and continuum filtergrams, Rimmele (2008) found that UDs, when observed in the G band, displayed dark fine-structuring toward their core. Contrarily, this author found that these dark features are not as readily visible in the continuum, and concluded this may be due to the reduced resolution of the longer wavelength and longer exposure images. However, our continuum images were obtained at a shorter wavelength than the G band (4170 Å versus 4305 Å), in addition to being captured with a much shorter exposure time (7 ms versus 20 ms). Thus, we cannot attribute a lack of dark fine-scale structuring in the UDs observed in the 4170 Å continuum to either reduced spatial resolution, or exposure-time-related smearing of the images. Jess et al. (2012) recently determined the contribution functions of the G-band and 4170 Å continuum filter bandpasses used at the DST, and concluded that the blue continuum is formed at a height of ∼25 km, while the G band is formed approximately 100 km higher. This atmospheric height separation between the two bandpasses, in addition to the appearance of diatomic CH molecules in the G-band images, may contribute to the presence of dark cores observed in G-band UDs.

Since we are concerned with the larger macro-scale fluctuations in umbral intensity, rather than the very small scale structuring (e.g., dark features within UDs) which may be close to, if not overlap with the telescope's diffraction limit, our photospheric umbral time series was created purely from consecutive 4170 Å continuum images. These images were multiplied by the umbral binary map, producing a data set which could be analyzed with Fourier techniques. Following the methodology of Jess et al. (2007a, 2007b), the wavelet analysis routines of Torrence & Compo (1998) were applied to the 4170 Å continuum umbral time series to search for the presence of oscillatory behavior. Considerable oscillatory power was present throughout the sunspot umbra, with dominant peaks in the Fourier spectrum at 3 and 5 minutes. This is consistent with the generalized p-mode spectrum of sunspot oscillations (Lites et al. 1982; Lites 1984, 1986; Bogdan & Judge 2006). However, when locations of high oscillatory power were examined, it became evident that the UDs displayed significant enhancements of wave amplitude, at periodicities of approximately 3 minutes. The middle panel of Figure 3 shows the spatial locations where oscillatory power at approximately 3 minutes is in excess of 1000 times the background umbral value. To test whether these long-lived umbral brightenings oscillate in phase with the surrounding umbra, temporally filtered time series of these structures were first created by passing each spatially averaged UD intensity through a 2–4 minute bandpass filter. The resulting light curves, dominated by power corresponding to the 3 minute p-mode oscillations, are displayed in the upper panel of Figure 4, along with a filtered time series representing a region of the quiet umbra devoid of any transient brightenings. It is clear that not only do the peaks and troughs of the UD oscillations appear similar in time, but these signatures also closely follow the oscillations originating from within the background umbra.

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To quantify any small-scale differences in the oscillating UD time series, a phase difference analysis with respect to the background umbra was performed. The lower panel of Figure 4 reveals the phase difference between each of the four UD time series, and an isolated region of the background umbra (i.e., the solid black line in the upper panel of Figure 4). It is clear that a preference exists for the UDs to oscillate in phase with the background umbra. Maximum deviations in the phase angle reach approximately ±40°, although these may be due to slight drifts in oscillation period between the respective time series. For example, the phase angle related to UD1 (red line in the lower panel of Figure 4) drifts a total of ≈80° over a 700 s time duration, providing an average shift of ≈011 s⁻¹. This drift can be explained by a period discrepancy of only 20 s between the UD and the background umbra, something which is well within the normal range of p-mode frequencies (Thomas 1985). Thus, we interpret the umbra as a single oscillating "drum skin," which not only induces wave motion in its inherent UDs, but also causes these structures to oscillate in phase with one another.

While Chou et al. (2009) and Stangalini et al. (2011) have shown that 3 minute magnetoacoustic power is suppressed in sunspot umbrae as a result of local absorption and emissivity reduction, their spatial resolution was insufficient to allow studies of the smallest sunspot features (e.g., UDs). Recently, Shelyag et al. (2009) have shown that on small spatial scales the curvature and strength of magnetic field lines have substantial influence on the efficiency of magnetoacoustic wave propagation. Indeed, these authors suggest that wave power can actually be amplified under certain atmospheric conditions. Following on from this, our observational data suggest that UDs are able to enhance the background p-mode power by at least three orders of magnitude (see, e.g., middle panel of Figure 3). This may be a result of the magnetic field lines acting as efficient conduits for magnetoacoustic wave propagation (e.g., Singh 1992; Shelyag et al. 2006; Khomenko et al. 2008; Erdélyi & Fedun 2010).

Chromospheric information comes from the Hα core imaging data set, in addition to the Ca ii 8542 Å spectral imaging scans taken with IBIS. Furthermore, Doppler shifts of the Ca ii profile minimum allow a series of two-dimensional velocity maps to be generated (Jess et al. 2010b). The Ca ii core image shown in the right panel of Figure 1 is a true intensity map, created by establishing the line-profile minimum at each pixel. By displaying Doppler-compensated line-center intensities, rather than rest-wavelength intensities, brightness variations throughout the image are more indicative of the source function than of the velocities present in the line-forming region (Leenaarts et al. 2010).

In a process identical to that applied to the photospheric image sequence, the Hα and Doppler-compensated Ca ii data sets were multiplied by the umbral binary map, and subsequently analyzed using wavelet techniques. As for the 4170 Å continuum time series, considerable oscillatory power was present throughout the entire sunspot umbra, with dominant peaks in the Fourier spectrum at 3 and 5 minutes. However, remaining consistent with Thomas (1985), chromospheric power at periodicities of ∼5 minutes was much reduced when compared to those found in the photospheric umbra. To investigate whether oscillatory power detected in the chromosphere can be related to similar phenomena found in the photosphere, the locations of high oscillatory power were examined. The right panel of Figure 3 outlines the spatial locations where power, in both intensity and velocity signals at approximately 3 minutes, is in excess of 1000 times the background umbral value. Similar to the locations of high photospheric power displayed in the middle panel of Figure 3, several distinct groups manifest in a horseshoe shape in the southwest quadrant of the sunspot umbra.

The locations of significant chromospheric power have two distinct differences when compared to their photospheric counterparts. First, their spatial positions are slightly offset from those displayed in the middle panel of Figure 3. Second, the spatial sizes of high chromospheric power are substantially larger than those found in the photosphere. These effects can be explained by the geometry of the magnetic field lines which stretch outward from the solar surface. For example, high photospheric power encompassing UD2 (middle panel of Figure 3) has a local maximum at the heliocentric coordinate (−1337, −4837), while the chromospheric local maximum is at (−1331, −4822). Thus, the offset between these two maxima is ≈16, or ≈1100 km. A separation between the 4170 Å continuum and the Hα/Ca ii core formation heights of ∼1800 km (Vernazza et al. 1981; Jess et al. 2012) requires an inclination angle of ≈40° to support the assumption that the oscillatory signals are from a continuation of the same solar structure (e.g., Centeno et al. 2006). This is the largest offset present in our observations, and forms our upper limit of the magnetic field inclination angle. Rempel et al. (2009) have recently shown that the inclination angles (to the vertical) of magnetic flux tubes within sunspot umbrae can be quite large (>45°), with these structures often becoming horizontal near the penumbral boundary. Furthermore, Marsh et al. (2009) utilized stereoscopic observations of propagating slow-mode waves in coronal structures to infer an absolute inclination angle ∼40°. Thus, an inclination angle of <40° supports the interpretation that oscillatory behavior detected at photospheric and chromospheric layers are directly related by the magnetic field lines which extend upward from the solar surface.

An increase in the area of the oscillating regions can be associated with a physical expansion of the magnetic flux tubes as a function of atmospheric height. Through examination of intense magnetic field concentrations (>1000 G), Jess et al. (2009) were able to show that the radial expansion of magnetic flux tubes between photospheric and chromospheric heights can be as large as a factor of two. Assuming these magnetic flux tubes demonstrate an expanding cylindrical geometry, doubling the radial dimension will result in an area increase of 400%. Continuing with the previous example, the region of high oscillatory power encompassing UD2 covers 326 pixels (0.8 Mm² at 2500 km² pixel⁻¹) and 256 pixels (2.5 Mm² at 10, 000 km² pixel⁻¹) at photospheric and chromospheric heights, respectively. As a result, an expansion of only 310% is observed, meaning our observations are well within the limits of previous expanding magnetic flux tube models (e.g., Mein et al. 2007; Ruderman et al. 2008; Shelyag et al. 2010; Karami & Bahari 2011; Fedun et al. 2011).

The specific mode of oscillation can be determined through investigation of the coupling between intensity and velocity signals. Here, we adopt the V − I convention (Deubner & Fleck 1989; Fleck & Deubner 1989), which shows the delay of maximum intensity (I) with respect to maximum blueshift velocity (V). Thus, a wave which has a velocity signal trailing its intensity signal by 1/4 of a period will have a V − I phase angle of −90°. Due to the Ca ii absorption line being sensitive to temperature fluctuations (Beebe & Johnson 1969), we follow the common practice to adopt the intensity (I) as a proxy for the local temperature. Mein (1977) has shown that when waves propagate along moderately inclined flux tubes, the phase lags between fluctuations in velocity and temperature are the same as those found in purely acoustic waves which have been modified by gravity. Therefore, in a fully adiabatic scenario, theory predicts a V − I phase angle approaching 0° for running acoustic waves, which can increase to −90° when aspects of wave reflection create standing acoustic modes (Hofmann et al. 1996; Al et al. 1998; Nigam & Kosovichev 1999). Under isothermal conditions, the phase lag can further increase up to −180° (Mihalas & Toomre 1981, 1982). Examination of the upper panel in Figure 5 reveals how the unfiltered velocity time series (dashed line) trails the co-spatial intensity light curve (solid line) by approximately 1/4 of a wave period. This effect becomes even more pronounced when the same time series are passed through a 2–4 minute bandpass filter (middle panel of Figure 5). Chromospheric waves detected within the contours of the right panel of Figure 3 have a spatially and temporally averaged phase angle of −87° ± 8°, suggesting these waves are magnetoacoustic in nature, with characteristics consistent with standing acoustic modes (Deubner 1974; Cram 1978). The generation of a chromospheric standing wave may be the result of a portion of the acoustic wave energy being reflected back at the transition region boundary (Schmitz & Fleck 1992; Nakariakov et al. 2004; Fedun et al. 2011).

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The spatial and temporal averaging of velocity signals found in the locations of high chromospheric power results in a net blueshift velocity of ≈1.5 km s⁻¹ (upper panel of Figure 5). While a V − I phase angle of −87° ± 8° implies the presence of wave reflection, a net blueshift velocity may also indicate that a portion of the magnetoacoustic wave is propagating in the upward direction. To examine the propagation characteristics of the magnetoacoustic waves associated with the UDs, spatially averaged Hα intensity time series were constructed for each of the regions contoured in the right panel of Figure 3, and subsequently passed through a 2–4 minute bandpass filter to isolate oscillatory phenomena around 3 minutes. A phase difference analysis was performed between these chromospheric Hα light curves and their corresponding photospheric 4170 Å continuum counterparts created in Section 3.1, and displayed in the upper panel of Figure 4. A preference for negative phase angles is shown in the lower panel of Figure 5, which indicates the oscillatory signals are first observed in the photospheric 4170 Å continuum, before later being detected in the chromospheric Hα time series. By averaging the derived phase angles over all UDs, an average phase lag of −43° ± 12° is found, implying the presence of upwardly propagating waves. Using the dominant periodicity of ≈3 minutes, a relative phase angle of −43° equates to a time lag of ≈22 s. This is consistent with the photosphere-to-chromosphere time-lag measurements detailed in Kobanov et al. (2011b). However, due to a height separation of ∼1800 km between the 4170 Å continuum and Hα core image sequences (Vernazza et al. 1981; Jess et al. 2012), a time lag of ≈22 s requires a phase speed exceeding 80 km s⁻¹. This is clearly unfeasible since a velocity this large will be supersonic, and violates our interpretation that the observed waves are magnetoacoustic slow modes. However, because the wave trains exist at the start of our observing sequence, and continue to be observed as the time series finishes, a factor of n2π (n360°), where n is an integer, may be absent from our derived phase angle. By considering n = 1, the phase lag increases to ≈−403°, with the resulting phase speed reducing considerably to ≈8 km s⁻¹. Only by observing the start and/or end of a propagating wave train can the exact time lag (and associated phase speed) be conclusively determined. Nevertheless, the detected waves are best described as upwardly propagating magnetoacoustic modes, which travel along magnetic flux tubes inclined to the vertical by less than ≈40°. Such an inclination angle may explain why Kobanov et al. (2008, 2011a) were unable to correlate (on a pixel-by-pixel basis) high chromospheric oscillatory power to that occurring in the underlying photosphere.

Our Hα data set has a much higher cadence than the Ca ii core image sequence (1.26 s instead of 43.4 s). Thus, examination of the Hα intensity time series can substantially reduce the associated errors when deriving the periodicities of the propagating magnetoacoustic waves. As can be seen in Figure 5, the Hα data display prominent intensity oscillations, most of which last for the entire duration of the time series. A spatial and temporal averaging over the regions encompassed by the red contours in the right panel of Figure 3 yields a peak chromospheric periodicity of 168 ± 7 s, consistent with the generalized p-mode spectrum of chromospheric sunspot oscillations (O'Shea et al. 2001, 2002; Banerjee et al. 2002). The close resemblance between Hα and Ca ii intensity time series suggests their formation heights are remarkably similar. This is also apparent by the similarities present in simultaneous snapshots through their respective filters (e.g., Figure 1).

Due to the reduced spatial resolution of AIA images (2 pixel resolution ≈12 ≈870 km), when compared with our simultaneous data sets of the lower solar atmosphere (2 pixel resolution ≈0138 ≈100 km), it is imperative to concentrate on the larger scale structures which will be apparent in all image sequences. Following inspection of the co-aligned images displayed in the lower panels of Figure 2, it is clear that a number of fan structures rise out of the photospheric sunspot umbra, and into the corona. These structures are particularly visible as intensity enhancements in the coronal AIA images, where the 131 Å, 171 Å, 193 Å, 211 Å, and 335 Å bandpasses show similar structuring extending outward in the southwest direction. However, these fan structures are mostly absent in the transition region 304 Å images.

To investigate whether an absence of the fan brightenings in the 304 Å images is a consequence of the structures lying outside of the filter's temperature response curve, differential emission measure (DEM) techniques were employed (e.g., Hannah & Kontar 2012). Utilizing the six coronal EUV channels on AIA, we were able to construct emission measure (EM) and temperature (T_e) estimates of active region NOAA 11250, including its immediate vicinity. Following the methodology presented by Aschwanden et al. (2011), each batch of near-simultaneous EUV exposures allowed us to construct a time series of EM and T_e variables, thus allowing the evolutionary changes in each sequence to be studied. Importantly, the temperature of the coronal fan structures currently under investigation is in the range of 0.5–1.2 MK (right panel of Figure 6). Their relatively cool temperature, when compared to the multi-million degree values found at the center of the active region, probably manifest as a result of open magnetic field lines which cannot trap heated plasma. An alternative explanation could revolve around their connection with quasi-separatrix layers, which may be subject to a peculiar heating function (Schrijver et al. 2010). While the 304 Å bandpass has two distinct temperature response functions covering approximately 50, 000 K and 1.5 MK, the fact that active region NOAA 11250 was positioned close to solar disk center suggests that the resulting images should be dominated by the He ii emission formed at ∼50, 000 K, with contributions from the 1.5 MK Si xi 202.22 Å emission line minimal (O'Dwyer et al. 2010). This helps to explain why features near one million degrees (e.g., the fan structures extending outward from the underlying sunspot), are not readily apparent in the transition region data.

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Examination of time-lapse movies of coronal EUV images revealed clear and distinctly periodic outflows along the coronal fans. To quantify the associated periodicities and flow velocities, a series of one-dimensional slits were placed along the motion path in each of the coronal EUV channels. The resulting time–distance cuts reveal numerous propagating wave fronts, indicated by straight, diagonal trends in the bottom panel of Figure 7. The AIA EUV images, in the temperature range of 0.4–2.8 MK (incorporating the 131 Å, 171 Å, 193 Å, 211 Å, and 335 Å bandpasses), display a dominant periodicity of 172 ± 17 s, and a propagation velocity of 45 ± 7 km s⁻¹. Red and blue contours in the lower panel of Figure 7 outline the intensity signals present in the higher temperature 211 Å and 335 Å bandpasses, respectively. It is clear that not only do the periodicities of the wave fronts closely resemble one another in both space and time, but the intensity gradients (and hence wave speeds) are also similar. Propagating wave fronts are observed co-spatially, and simultaneously in all but the 94 Å AIA bandpass. The signal to noise of the 94 Å channel is too low to extract a time series of sufficient quality for analysis. This may be compounded by the fact that the fan structures demonstrate a temperature much lower than the 94 Å channel's peak response.

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A periodicity of ≈172 s and a propagation velocity ≈45 km s⁻¹ are consistent with King et al. (2003), who utilized the 171 Å and 195 Å filters onboard the TRACE satellite to reveal how the propagation velocity of such waves will not exceed the local sound speed, even with a large inclination angle of the waveguides away from the observer's line of sight. The typical sound speed associated with the dominant Fe ix emission from the 171 Å bandpass is ≈150 km s⁻¹ (De Moortel 2006), which requires the fans currently under investigation to have an inclination angle exceeding 70° before the observed wave motion would become supersonic. An inclination angle this large is highly unrealistic due to the location of the active region on the solar disk, in addition to previous surveys of coronal magnetic geometries (e.g., Aschwanden et al. 2002, 2008, 2009). Thus, we can conclude that the observed wave phenomenon is best described as propagating slow-mode waves.

Since propagating wave fronts are detected in many of the AIA EUV bandpasses, EM and temperature maps were subsequently investigated for the presence of similar signatures. As the EM maps created here are the sum of (squared) electron densities along a given line of sight, the total mass distribution, regardless of the local temperature, can be studied as a function of time (Aschwanden et al. 2011). Furthermore, the derived pixel-by-pixel temperatures correspond to the peak of the DEM distribution for a given line of sight, thus allowing the entire temperature range of the coronal plasma to be easily studied (Aschwanden & Boerner 2011). Applying the same techniques used on the EUV image sequences, time–distance diagrams of the EM and T_e variables derived for the fan structures reveal identical periodicities and propagation velocities as found in the EUV images (middle and upper panels of Figure 8). The EM is usually defined as (Kingston et al. 1982; Doyle et al. 1985)

$$\begin{equation} {\rm EM} (T_{e}) = \int n_{e}^{2} dh, \end{equation} \tag{ 1 }$$

where n_e is the electron density and h is an emitting depth along the observer's line of sight. Since the compressive phase of a magnetoacoustic wave mode will cause the emitting volume to decrease, thus reducing the overall EM, one would expect the EM time series to oscillate in phase with the corresponding EUV intensities. However, as the plasma becomes compressed, the associated kinetic temperature will increase, thus causing the T_e signal to oscillate out of phase with the EM and EUV intensity signals. Indeed, perturbations in the EUV intensity, depicted in Figure 7, are found to occur in phase with the derived EM fluctuations. Furthermore, the lower panel in Figure 8 reveals how oscillations in T_e are found to be 180° out of phase with respect to those detected in the EM signal. Quantitatively, the detected fluctuations in the EM and T_e values are typically in the range 21.50  ±  0.03 cm⁻⁵ K⁻¹ and 0.55  ±  0.03 MK, respectively. This strengthens our interpretation that the observed wave phenomena are best described as magnetoacoustic slow modes.

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To model our observational findings, we performed a series of one-dimensional numerical simulations using the Lagrangian-Remap code (lare; Arber et al. 2001), in which thermal conduction and optically thin radiation have been included (Owen & De Moortel 2009). The one-dimensional model setup is constructed to closely match the observed AIA 171 Å emission at the lowest coronal part of the fan structure. In particular, the density profile (upper-left panel of Figure 9) along the fan includes gravitational stratification, where the initial (z = 0) value is determined through equalization of the forward-modeled and observed 171 Å intensities, as can be seen in the upper-right panel of Figure 9. Here, the solid line corresponds to the spatial variation in 171 Å emission, obtained through forward-modeling of the temperature and density profiles inferred from the observed fan structure (where the forward modeling is undertaken using a code developed by De Moortel & Bradshaw 2008). A model temperature profile is constructed which utilizes the minimum and maximum values of the (isothermal) temperatures extracted directly from the AIA emission. The background magnetic field is taken to be constant. From the upper-right panel of Figure 9, it is clear that the theoretical model is in close agreement with the observed 171 Å emission during the first 15 Mm along the fan. After this, the model emission declines more steeply than the observed 171 Å emission. However, we are primarily concerned with the behavior within the first 15 Mm, and therefore can neglect larger distances where the emission becomes faint, and the observations become more affected by detector noise.

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A slow magnetoacoustic wave is driven into the domain at the lower boundary, through periodic perturbations of the field-aligned velocity component. A snapshot of the velocity perturbations at t = 30 minutes, using a driving period of 172 s to match that of the observations, is shown in the lower-left panel of Figure 9. A slow wave will propagate along the field at the characteristic tube speed, which is of the order of the local sound speed. Incorporating a temperature ≈1 MK, and a background magnetic field strength ≈10 G, the tube speed is of the order of 122 km s⁻¹. Combined with a driving period of 172 s, we expect a wavelength of the order of 22 Mm, which can be verified in the lower panels of Figure 9. After an initial increase, due to the rapidly decreasing background density, the velocity amplitudes are strongly damped through a combination of thermal conduction, optically thin radiation, and compressive viscosity. The corresponding density perturbations (i.e., the density profile at t = 0 has been subtracted from the density at t = 30 minutes) are shown in the lower-right panel of Figure 9. Such strong damping is predominantly caused by thermal conduction, as was previously uncovered by De Moortel & Hood (2003, 2004) and Owen & De Moortel (2009). Only a small fraction of the overall damping can be attributed to optically thin radiation and compressive viscosity.

To achieve a more meaningful comparison with the perturbations observed by the AIA instrument, we utilized the model density and energy (temperature) distributions along the loop, to forward-model the emission for different AIA channels, at a multitude of time steps. We find that the modeled 171 Å emission is significantly more intense over the first 15 Mm of the coronal fan structure, when compared to the 193 Å and 211 Å intensities. Although these AIA channels contain some emission lines at cooler temperatures, the 193 Å and 211 Å bandpasses mainly respond to hotter coronal temperatures (i.e., above 1.5 MK; O'Dwyer et al. 2010; Brooks et al. 2011), and hence only begin to display elevated intensities when the density reduces and the local temperature increases beyond 1 MK. This is consistent with the AIA observations, whereby the cooler 171 Å emission dominates the first 15 Mm from the underlying sunspot. When forward-modeling intensity oscillations, it is important to remember that a variation in amplitude of the intensity perturbations is a combination of the change in amplitude of the model density and temperature fluctuations, as well as the individual response functions of the AIA channels to different temperatures.

Finally, we utilize the global wavelet power, at each spatial position along the coronal fan, to compare the amplitude decay rates found in the different AIA bandpasses in a more quantitative way. To do this, a wavelet transform was performed at each position along the fan, with the temporally averaged global wavelet power subsequently calculated. The red, green, and blue lines in Figure 10 represent the time-averaged oscillatory power around 3 minutes, for the AIA 171 Å, 193 Å, and 211 Å observed and forward-modeled bandpasses, respectively. Note that each of these has been normalized to their individual maxima for display purposes. Of particular note is the observed 193 Å power spectrum. Here, the solid green line represents the oscillatory power registered through the entire AIA bandpass, while the dashed green lines relate to the power associated with the hot (∼1.6 MK) and cool (≲1 MK) spectral components of this channel. These components have been separated using the methods detailed in Kiddie et al. (2012), with the resulting cool emission peaking alongside the 171 Å power at ≈4.5 Mm, and the hot emission displaying a peak between the 171 Å and 211 Å bandpasses (≈3 Mm). In reality, the global wavelet power of the 193 Å and 211 Å emission is substantially lower than that of the 171 Å emission. However, the time-averaged power of the 193 Å and 211 Å modeled intensities are very similar, as is the case for the observations. For these wavelengths, the simulated global wavelet power reaches a minimum around 10 Mm, while the 171 Å bandpass reaches its minimum power slightly further along the fan structure, at about 15 Mm. These results agree qualitatively with the observed time-averaged power, where the 193 Å and 211 Å channels near-simultaneously reach a minimum at about 9 Mm along the coronal fan, with the 171 Å power diminishing completely by 15 Mm. This reiterates the importance of thermal conduction in the damping of coronal slow-mode waves. The increase in time-averaged global wavelet power, after ≈10 Mm in both the simulated and observed 193 Å and 211 Å time series, is due to the fact that, despite the decreasing amplitudes of the model density and temperature, the percentage intensity perturbations in these channels are actually increasing. This is a result of the increased sensitivity of the AIA 193 Å and 211 Å bandpasses to the >1 MK temperatures which have been reached by this distance along the fan structure. The increase is not as apparent in the observed 193 Å and 211 Å time-averaged wavelet power spectra, which may be due to the decreased signal-to-noise levels found in these channels, especially when compared to the 171 Å bandpass.

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