Activityrelated variations of highdegree pmode amplitude, width, and energy in solar active regions
^{1} Department of Physics and Astronomy, Seoul National University, 151747 Seoul, Republic of Korea
email: ramajor@astro.snu.ac.kr; jcchae@snu.ac.kr
^{2} Udaipur Solar Observatory, Physical Research Laboratory, 313001 Udaipur, India
email: ambastha@prl.res.in
Received: 6 February 2013
Accepted: 14 November 2013
Context. Solar energetic transients such as flares and coronal mass ejections occur mostly within active regions (ARs) and release large amounts of energy, which is expected to excite acoustic waves by transferring the mechanical impulse of the thermal expansion of the flare on the photosphere. On the other hand, strong magnetic fields of AR sunspots absorb the power of the photospheric oscillation modes.
Aims. We study the properties of highdegree pmode oscillations in flaring and dormant ARs and compare them with those in corresponding quiet regions (QRs) to find the association of the mode parameters with magnetic and flarerelated activities.
Methods. We computed the mode parameters using the ringdiagram technique. The magneticactivity indices (MAIs) of ARs and QRs were determined from the lineofsight magnetograms. The flare indices (FIs) of ARs were obtained from the GOES Xray fluxes. Mode parameters were corrected for foreshortening, duty cycle, and MAI using multiple nonlinear regression.
Results. Our analysis of several flaring and dormant ARs observed during the Carrington rotations 1980–2109 showed a strong association of the mode amplitude, width, and energy with magnetic and flare activities, although their changes are combined effects of foreshortening, duty cycle, magneticactivity, flareactivity, and measurement uncertainties. We find that the largest reduction in mode amplitude and background power of an AR are caused by the angular distance of the AR from the solar disc centre. After correcting the mode parameters for foreshortening and duty cycle, we find that the mode amplitudes of flaring and dormant ARs are lower than in corresponding QRs reducing with increasing MAI, suggesting a stronger mode power suppression in ARs with larger magnetic fields. The mode widths in ARs are larger than in corresponding QRs and increase with MAI, indicating shorter lifetimes of modes in ARs than in QRs. The variations in mode amplitude and width with MAI are not same in different frequency bands. The largest amplification (reduction) in mode amplitude (mode width) of dormant ARs is found in the fiveminute frequency band. The average mode energy of both the flaring and dormant ARs is smaller than in their corresponding QRs, reducing with increasing MAI. But the average mode energy reduction rate in flaring ARs is smaller than in dormant ARs. Moreover, the increase in mode width rate in dormant (flaring) ARs is followed by a decrease (increase) in the amplitude variation rate. Furthermore, including the mode corrections for MAI shows that mode amplitude and mode energy of flaring ARs escalate with FI, while the mode width shows an opposite trend, suggesting excitations of modes and growth in their lifetimes by flares. The increase (decrease) in mode amplitude (width) is larger in the fiveminute and higherfrequency bands. The enhancement in width variation rate is followed by a rapid decline in the amplitude variation rate.
Key words: Sun: helioseismology / Sun: magnetic field / Sun: activity / Sun: flares
© ESO, 2014
1. Introduction
Photospheric fiveminute oscillations, probably first observed by Leighton et al. (1962), are caused by trapped acoustic waves (pmodes) inside the solar interior (Ulrich 1970; Leibacher & Stein 1971) and are well known and have been studied extensively. It is believed that the energy of pmodes is contributed by convective or radiative fluxes. A precise determination of the pmodes properties provides a powerful tool to probe the solar interior. Highdegree (ℓ > 200) acoustic oscillations are vertically trapped in a spherical shell with the photosphere as the upper boundary and the lower boundary depending on the horizontal wavenumber, , and the frequency (ω), (1)where r_{t} is depth of the lower turning point. Lifetimes of highdegree modes are much shorter than the sound travel time around the Sun, therefore local effects are more important for these modes than for the lowdegree modes, which have longer horizontal wavelengths and longer lifetimes. It is likely that highdegree acoustic waves are not global modes, that is, they do not remain coherent while travelling over the circumference to interfere with themselves. Therefore, they can locally be considered as horizontally travelling, vertically trapped waves. These are observed as photospheric motions inferred from the Doppler shifts of photospheric spectral lines. The analysis of local modes provides a diagnostic tool to study the structural and dynamic properties of the solar interior beneath ARs as well as the changes in excitation and damping of modes that are most affected by the surface magnetic fields.
Previous studies showed that characteristics of the highdegree modes are modified in active regions (ARs) with complex and strong magnetic fields associated with sunspots, which are different from magnetically quiet regions (QRs). The significant reduction of the pmode energy in ARs has been attributed to absorption by sunspots (Braun et al. 1987; Braun & Duvall 1990; Bogdan et al. 1993; Zhang 1997; Hindman & Brown 1998; Haber et al. 1999; Mathew 2008; Gosain et al. 2011). Previous studies showed that the mode power absorption changes with harmonic degree and radial order. The absorption increases with increasing horizontal wavenumber k_{h} over the range 0.0–0.8 Mm^{1} and decreases for higher k_{h} in the range 0.8–1.5 Mm^{1}. The absorption along each individual pmode ridge tends to peak at an intermediate value of the degree 200–400. Hindman & Brown (1998) found that the amplitudes of oscillations with frequencies lower than 5.2 mHz decrease with field strength for both velocity and continuum intensity measurements. Furthermore, they reported that oscillations with frequencies between 5.2 and 7.0 mHz within ARs suppressed continuum intensity amplitudes, but enhanced velocity amplitudes.
The differences in mode frequency and width in ARs and QRs have been studied earlier. Rajaguru et al. (2001) found that the frequencies of solar oscillations are significantly higher in ARs than in QRs. Width and asymmetry of the peaks in power spectra are also larger in ARs, while the mode amplitudes are smaller. Furthermore, Howe et al. (2004) showed that the difference in mode characteristics are correlated with the differences in the average surface magnetic fields between corresponding regions. They reported a strong dependence of amplitude and lifetime of pmodes on the local magnetic flux, and found that these parameters decreased in the fiveminute band, while a reverse trend was found at high frequencies. RabelloSoares et al. (2008) confirmed previous results and reported that mode amplitude and width variations are nearly linear.
Changes in the mode parameters with solar activity cycle have also been studied by many researchers. Tripathy et al. (2010) found that changes in mode frequency during the activityminimum period are significantly stronger than during the solar maximum. Komm et al. (2000a) reported a 23% increase in mode widths (i.e., decrease in mode lifetimes) with increasing solar activity. They found that the variations are frequency dependent; they are strongest near 3.1 mHz, and are independent of harmonic degree ℓ. Changes in mode amplitudes, power, and energysupply rate were analysed from solar minimum to maximum. It was found that mode amplitudes and solar activity level are anticorrelated for intermediate and highdegree modes, and strongly depend on local magnetic fields. Chaplin et al. (2000) suggested that the changes in mode parameters arise from changes in damping instead of excitation. Burtseva et al. (2009a) found that the mode lifetime in ARs and QRs decreases with magneticactivity, but the decrease is slower in QRs. Moreover, pmode amplitudes at solar minimum are higher than at solar maximum (Burtseva et al. 2009b).
Energetic transients, viz., flares, and coronal mass ejections (CMEs), are believed to be caused by reconnection of magnetic fieldlines in the solar atmosphere. Particles are energized at the primary energyrelease site in the corona and then guided along the magnetic fieldlines downwards to the denser atmosphere. They are expected to affect and excite the highdegree pmodes by transferring mechanical impulse of thermal expansion on the photospheric layer (Wolff 1972). The observational evidence of flareinduced horizontally travelling waves on the solar photosphere has been reported previously (Kosovichev & Zharkova 1998; Donea et al. 1999; Donea & Lindsey 2005; Kosovichev 2006, 2011). The detection of seismic waves provides us with a unique opportunity to study the excitation of solar oscillations, and raise new questions about the underlying physical processes as well as the properties of the excited waves and source producing them. But the observations of seismic waves are relatively rare, possibly because of the difficulties of detecting the photospheric ripples. Such travelling waves appear to be associated mostly with energetic flares.
Flarerelated variations in highℓ pmode parameters have been studied previously (Ambastha et al. 2003; Maurya et al. 2009; Maurya 2010; Maurya & Ambastha 2011). Ambastha et al. (2003) have found that power in highdegree pmodes is stronger amplified during periods of high flareactivity in some ARs than in the nonflaring ARs of similar magnetic fields. Maurya et al. (2009) found strong evidence of a substantial increase in mode amplitude during an extremely energetic flare. On the other hand, at a global scale of the whole solar disc, Ambastha & Antia (2006) found a poor correlation between the running mean of FI and lowℓ mode power. They reported similar changes for the CME index. Karoff & Kjeldsen (2008) found that energy in the flare can drive global oscillations in the Sun in the same way as the Earth is set ringing after a major earthquake. They reported a stronger correlation between flares and oscillation energy at high frequencies than for the ordinary pmodes. Flareinduced excitations at higher frequencies were also reported by Kumar et al. (2010, 2011). A more recent study by Richardson et al. (2012), however, did not find evidence of flaredriven highfrequency global modes. These highfrequency modes, unlike the pmodes, are not expected to be trapped in the solar interior.
Solar ARs, where the flares typically occur, are locations of strong magnetic fields associated with the sunspots. Thus, there is a competition between the absorption of pmode energy by sunspots and excitation/amplification by the energetic transients. The net increase or decrease of the mode energy would, therefore, depend on their relative dominance. Characteristics of the photospheric oscillations in ARs are essentially described by the modes of shorter wavelength, which are trapped just below the photosphere. These modes can be studied using the socalled ringdiagram analysis (Hill 1988), which is a wellestablished technique in local helioseismology. It is a generalization of the globalmode analysis for small patches of the solar surface, where the geometry is approximately plane parallel, and waves can be treated as plane waves. The modes measured using this technique are typically ℓ > 150, which have a lower turning depth above r/R_{⊙} ≈ 0.95. For more details about this technique, we refer to the review by Antia & Basu (2007).
The ringdiagram technique uses several hourslong timeseries of Doppler observation which may dilute the shorterduration flareinduced effects owing to the averaging involved in space and time. Nevertheless, we found strong evidence of flarerelated amplification in pmode amplitude during the longduration, high magnitude X17.2/4B flare of 28 October 2003 (Maurya et al. 2009). This study illustrated the possibility of detecting changes in pmode parameters for longduration energetic flares with a ringdiagram analysis. The observed amplification in pmode amplitude implies that during an energetic flare, the energy of pmodes must be weighted by the energy of excited modes. To further establish these results, we investigate several ARs, both flare producing and dormant along with QRs. We undertake a set of 53 flaring ARs of varying magnetic complexity to examine the relationship of mode amplitude, width, and energy with magnetic fields and flare energies. However, the ringdiagram analysis can also be used to study the frequency shift, subphotospheric flow, sound speed, etc.
Fig. 1 Distribution of flaring (filled circle) and dormant (unfilled circle) ARs used in this study as shown in a Carrington map. Circle sizes correspond to the magneticactivity index (B; for detail see Sect. 3.3) of ARs in Gauss. 

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The paper is organized as follows: Sect. 2 describes the selected sample of flaring and dormant ARs and the observational data used for our study. The methods of data analysis and the results are given in Sect. 3. We briefly discuss how the mode parameters are affected by foreshortening, duty cycles, and magnetic activities, and present an empirical method to correct them. We assumed that these effects are the only systematic variations and that measurement uncertainties are essentially random and are compensated for by averaging over many data sets. In Sect. 4 we present the results of the analysis and discuss them. We study variations in mode parameters with magnetic and flareactivity indices of ARs. We present average properties of the mode parameters and not their temporal variations. Section 5 gives the summary and conclusions.
2. Active regions and observational data
2.1. Flaring and dormant active regions
We selected our sample of energetic events occurring in different ARs using the archived information on ARs and solar activity from the webpages of the Solar Monitor^{1}. We first identified flares of Xray classes >M5.0 during the period of Carrington rotations 1980–2109 of solar cycles 23 and 24. We shortlisted the ARs lying within the angular distance of 40° from the solar disc centre and selected 53 events that were well covered by the Global Oscillation Network Group (GONG) project. Out of 53 events, 48 events correspond to solar cycle 23 and 5 to the current solar cycle 24. We also selected suitable QRs corresponding to every flaring AR located at the same latitude but at a different Carrington longitude.
To study the flareinduced excitation in flaring ARs as compared with nonflaring ARs, we also selected dormant ARs and associated QRs as for the flaring ARs. Thus, for every flare event, we selected four regions: flaring and dormant ARs and their associated QRs for the same time period. The locations and distribution of flare events and corresponding ARs over the solar disc are illustrated in Fig. 1. This shows that most of the events selected in our sample occurred within the latitude zone of ± 20^{o}. For brevity, throughout we used the notation B for the magneticactivity index (MAI) of an AR, measured in Gauss, which is not to be confused with the magneticfield strength.
2.2. Observational data
The observational data used in this study consist of merged Dopplergrams at oneminute cadence, obtained by GONG (Harvey et al. 1988), 96min averaged magnetograms from the Michelson Doppler Imager (MDI, Scherrer et al. 1995) on board the Solar and Heliospheric Observatory (SOHO), and the Xray fluxes from the GOES data archive. The pixel resolutions for GONG Dopplergrams and MDI magnetograms are 2.5′′ and 2.0′′, respectively. The 96min averaged magnetograms provided by MDI data archive are averages of calibrated oneminute magnetograms (with a noise level of ~30 G) and fiveminute magnetograms (with noise level of ~15 G).
3. Analysis techniques
3.1. Ring diagrams and pmode parameters
To estimate the pmode parameters corresponding to a selected area over the Sun, the region of interest is tracked over time. This spatiotemporal area is defined by an array (or data cube) of dimension N_{x} × N_{y} × N_{t}. Here, first two dimension (N_{x},N_{y}) correspond to the spatial size of the AR along x and yaxes, representing zonal and meridional directions, and the third (N_{t}) to the time t in minutes. The data cubes employed for the ring diagram analysis have typically duration of 1664 min and cover area of 16° × 16° centred around the location of interest. This choice of area is a compromise between the spatial resolution on the Sun, the range of depth and the resolution in spatial wavenumber of the power spectra. A larger size allows accessing the deeper subphotospheric layers, but only with a coarser spatial resolution. On the other hand, a smaller size not only limits access to the deeper layers, but also renders the fitting of rings more difficult.
The spatial coordinates of pixels in tracked images are not always integer. To apply the threedimensional Fourier transform on tracked data cube, we interpolated the coordinates of tracked images to integer values, for which we use the sinc interpolation method. Threedimensional Fourier transformation of data cube truncates the rings near the edges due to the aliasing of higher frequencies toward lower side. To avoid the truncation effects, we apodized the data cube in both the spatial and temporal dimensions. The spatial apodization was obtained by a 2Dcosine bell method, which reduces the 16° × 16° area to a circular patch with a radius of 15° (Corbard et al. 2003).
The observed photospheric velocity signal v(x,y,t) in the data cube is a function of position (x,y) and time (t). Let the velocity signal in frequency domain be f(k_{x},k_{y},ω), where, k_{x} and k_{y} are spatial frequencies in x and y directions, respectively, and ω is the angular frequency of oscillations. Then the data cube v(x,y,t) can be written as (2)The amplitude f(k_{x},k_{y},ω) of pmode oscillations is calculated using threedimensional Fourier transformation of Eq. (2). The power spectrum is given by (3)The 16^{o} patch from GONG consists of 128 pixels, giving a spatial resolution Δx =1.5184 Mm, that is, the knumber resolution, Δk = 3.2328 × 10^{2} Mm^{1}, and Nyquist value and resolution for the harmonic degree (ℓ), approximately 1440 and 22.5, respectively. The corresponding range in k_{x}k_{y} space is (− 2.069, 2.069) Mm^{1}. The temporal cadence and duration of the datasets give a Nyquist frequency of 8333 μHz and a frequency resolution of 10 μHz, respectively.
To determine various pmode parameters, the threedimensional power spectrum is fitted using the Lorentzian function (Haber et al. 2002; Hill et al. 2003) given by (4)where, is the total horizontal wavenumber. It can be identified with the degree (ℓ) of a spherical harmonic mode of global oscillations by . The six fitting parameters, viz., zonal velocity (U_{x}), meridional velocity (U_{y}), background power (b_{0}), mode central frequency (ω_{0}), mode width (Γ), and the parameter A′ are determined by the maximumlikelihood approach (Anderson et al. 1990). Note that A′ is not the amplitude (or mode height); the amplitude (A) can be determined from the expression, A′ = A × Γ^{2}. For convenience we use the term “mode” to refer to these parameters, although they are not discrete modes in the same sense as those observed at lower degrees. We are here mainly interested in the flare and magneticactivity associated changes in mode parameters: A, b_{0}, Γ, and mode energy (see Sect. 3.2).
A sample of pmode parameters computed using the above method is shown in Fig. 2 for AR NOAA 10649 obtained for the data cube on 17 July 2004. Panel a shows the wellknown ℓ − ν diagram, which represents the dispersion relation of the acoustic waves trapped in the subphotospheric depths of the AR. Different ridges correspond to modes of different radial orders, n = 0,...,5. Modes with higher harmonic degree ℓ correspond to shorter wavelengths, which are trapped in shallower depths (see Eq. (1)). These modes are more likely to be modified by the near surface activities, such as flares and sunspotmagnetic fields. Other panels show the natural logarithm values of the mode parameters, A′, mode width (Γ), and background power (b_{0}) as a function of frequency.
Fig. 2 pmode parameters obtained for AR NOAA 10649 on 17 July 2004: a) harmonic degree ℓ; b) mode amplitude A; c) mode width Γ; d) background power b_{0}; and e) mode area A × Γ as a function of frequency for radial orders n = 0,...,5. 

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3.2. The pmode energy
Photospheric pmode parameters provide a diagnostic tool to study subphotospheric structures and dynamics of ARs. For instance, mode width and amplitude provide clues to the excitation and damping mechanisms of solar oscillations. The line width (Γ) is directly related to the lifetime, (2πΓ)^{1}, of pmodes, and amplitudes can be converted into power and energy per mode. Total energy (i.e., kinetic and potential energy) of the p mode can be given by (Goldreich & Murray 1994) (5)where the parameter M_{n.ℓ} represents the mode mass, which can be computed using the mode inertia, and is mean square velocity, the measure of the mode area. It is given by, (6)where A_{nℓ} and Γ_{nℓ} are mode amplitude and mode width, respectively, obtained from the ring fitting (see Eq. (4)). This expression for the pmode energy (Eq. (5)) has been used previously to study solar cyclic variation of the pmode energy of globalmodes e.g., Komm et al. (2000b). The expression for the mean square velocity is different from the expression given in Komm et al. (2000b), who used the correction factor C_{vis} = 3.33 for the reduced visibility due to leakage (Hill & Howe 1998). But for the local modes, C_{vis}=1. We also note that the mode widths computed with the ringdiagram analysis are larger than the actual values and hence the estimated mode energy will be higher. Furthermore, there are many systematic effects on pmode parameters obtained with the ringdiagram analysis that make it difficult to determine the absolute mode parameters of a given region. Most of these systematic effects can be eliminated, however, by studying the differences in the fitted parameters between different regions with the same observing geometry. Therefore, we analyse mode parameters in ARs and corresponding QRs at same latitude, but at different Carrington longitude. The relative value of the mode energy, using Eqs. (5) and (6), between an AR and corresponding QR is given by (7)This expression for the mode energy is free from mode mass, and hence we do not require mode inertia. Similarly, we analysed fractional differences for other parameters, that is, mode amplitude (δA/A), mode width (δΓ/Γ), and background power (δb_{0}/b_{0}) of flaring and dormant ARs along with their corresponding QRs.
3.3. Magnetic and flareactivity indices
The MAI (denoted as B) for every data set of ARs and QRs was calculated from the 96 min averaged MDI magnetograms. Every magnetogram was tracked and remapped in the same manner as the Dopplergrams for the ringdiagram analysis. Then areas (16° × 16°) of the ring patches were extracted from the fulldisc images. Then, the MAI, B, was computed by taking the absolute average over the patches. Since the MDI 96 min averaged magnetograms were obtained from both one and fiveminute magnetograms, we set zero to the pixels with values lower than 35 G. Mathematically, the MAI of an AR can be written as (8)where b_{1} is the lowest threshold (=35 G), N_{1} is the number of pixels with  b  > b_{1}, N_{2} is the number of magnetograms observed during the ringdiagram data cube. Thus, the values of B for our samples of flaring and dormant ARs varies in the ranges of 37–308 G and 15–90 G, respectively, while for the QRs it ranges between 1–12 G.
Fig. 3 Relation between magneticactivity index (B) and flareactivity index (FI) of flaring ARs. Filled circles represent flaring ARs of our sample. The solid line shows the linear regression line, while the dashed lines around it correspond to the 95% confidence levels of the linear fit. 

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A measure of the flareactivity of an AR, the Xray flare index (say, FI), was calculated by multiplying the GOES Xray flux with the flare duration and then summing the contributions of all the flares that occurred during the 1664 min of the data cube for the given AR. The flare index of an AR thus can be written as (9)where F_{i} is the GOES Xray flux for i^{th} flare event, δt_{i} is the time duration of the same event, and N is the total number of flare events that occurred in the time period of the data cube. GOES Xray fluxes for different flare classes are listed in Table 1. Note that here we only considered the flares of C, M, and X classes.
GOES Xclass flares and associated energy fluxes.
Thus the flareactivity index computed for the flaring ARs is found to range from 8.9 × 10^{2} for moderately active ARs to 1.6 × 10^{5} erg cm^{2} for very productive ARs. Figure 3 shows the relation between the magneticactivity and flareactivity indices of our sample of flaring ARs. The straight line fit through the data points shows that the flaringactivity index increases with the MAI of ARs. The 95% confidence levels (dotted curves) of the fitting further validate this linear relation between FI and B.
Fig. 4 Mode parameters, amplitude A (top row), background power b_{0} (middle row), and width Γ (bottom row) for two modes (n = 0,ℓ = 652 (left three columns); n = 1,ℓ = 449 (right three columns)) for dormant ARs and QRs as a function of the central meridian distance λ of the region centres (left), duty cycle f (middle), and magneticactivity index B (right). Dashed curves show the quadratic fit (in λ) and the linear fit (in f and B) through the mode parameters. 

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Fig. 5 Multiple nonlinear regression coefficients, C_{1} (top row), C_{2} (middle row), and C_{9} (bottom row), normalized with intercept, C_{0}, for the mode parameters, A (left column), Γ (middle column), and b_{0} (right column) as a function of harmonic degree, ℓ. Corresponding correlation coefficients are shown in Fig. 6. 

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Fig. 6 Correlation coefficients, R_{1} (top row), R_{2} (middle row), and R_{9} (bottom row) corresponding to first and secondorder terms in central meridian distance λ (see Eq. (10)), respectively, for the mode parameters, A (left column), Γ (middle column), and b_{0} (right column) as a function of harmonic degree, ℓ. 

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3.4. Mode corrections
For a given multiplet (n, ℓ), mode parameters change from region to region and with time. These changes are the combined effects of foreshortening, duty cycle, magneticactivity, flareactivity, and measurement uncertainties. To study the activityrelated changes in oscillation modes, we need to analyse and correct for the other effects.
Here the duty cycle is the filling of the observed Dopplergrams in the data cube used for pmode computation with the ringdiagram analysis. In our data samples, the duty cycle varies in the range 70–100%. However, most of the data cubes have duty cycles >80%. Note that we analysed the relative variations in the mode parameters of ARs and corresponding QRs (e.g., Eq. (7)) that have the same duty cycles.
The Dopplergrams are significantly affected by foreshortening effects, i.e., as we go away from the disc centre, we measure only the cosine component of the vertical displacement. Another effect of foreshortening in mode parameters is caused by the fact that the spatial resolution in Dopplergrams decreases as we observe increasingly towards the limb. For example, a pixel which has a spatial resolution of dx at the disc centre, now images a horizontal distance on the Sun of dx/ρ, where ρ = cosΛsinΘ. This reduces the spatial resolution measured on the Sun in the centretolimb direction, and hence leads to systematic errors.
Since we aim to compare the mode parameters in different ARs, we analysed only the common modes among them. A mode is called common if all the data sets of ARs and QRs have same radial order (n) and degree (ℓ). We found a total of 98 common modes in our sample of ARs and QRs, while the total number of fitted modes ranges between 125–173. Figure 4 shows examples of changes in common modeparameters of all the dormant ARs and QRs with the central meridian distance (λ), the duty cycle (f), and the MAI (B).
One can infer from Fig. 4 that the mode amplitude (A) and background power (b_{0}) are more affected with distance from the disc centre than the mode width (Γ). The mode amplitude and background power decrease with increasing distance from the disc center, while the mode width shows an opposite relation. The variations in mode parameters with λ are best modelled by the quadratic function in λ (see the quadratic fit in Fig. 4). The variations in background power with distance from the disc centre are stronger than the changes with the MAI. We also note that the measured longitude and latitude are affected by the position angle (P angle) and B_{0} angle of the Sun.
Changes in mode parameters with duty cycles are shown in Fig. 4 (middle columns). The mode amplitude increases with increasing duty cycle in the data cubes, while the mode width and background power show the opposite trend. However, it is evident that the dutycycle related variations in mode parameters are much weaker than the variations due to the foreshortening and magneticactivity indices.
We modelled the effects of foreshortening and duty cycles on a common modeparameter () of all the ARs and corresponding QRs by a multiple nonlinear regression as a function of the central meridian distance (λ) and latitude (θ) of the centres of ring patches, position angle (P), and Bangle (B_{0}) of the Sun, and the duty cycle (f) of data cubes, (10)where C_{0}, C_{1}, ..., C_{9} are regression coefficients. We repeated this analysis for all the common pairs (n,ℓ) of the ARs and QRs.
Fig. 7 Average common modeparameters in QRs (black), dormant ARs (DARs; blue), and flaring ARs (FARs; red) for different radial orders as a function of mode frequency. Note that mode parameters are corrected for foreshortening and duty cycle before averaging. (This figure is available in color in electronic form.) 

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3.4.1. Effects of the foreshortening on the mode parameters
Figure 5 shows the multiple nonlinear regression coefficients (C_{1}, C_{2} and C_{9}), normalized by the intercept C_{0}, for the common modes in all ARs and QRs as a function of the harmonic degree, ℓ. The corresponding Pearson correlation coefficients are shown in Fig. 6. The coefficients C_{2} of the secondorder term for all the mode parameters are lower than the coefficients C_{1} of the firstorder term by one order of magnitude. But the correlation coefficients R_{2} for all mode parameters are twice as high as the coefficients R_{1}, suggesting a better association between the mode parameters and the quadratic term in λ. Two coefficients C_{1} and C_{2} are negative for the mode parameters A and b_{0} for all the harmonic degrees. For the mode width Γ, values of C_{1} are positive for most of the modes with radial order n = 0, 1 and negative for n = 2, 3. Moreover, the magnitude of the coefficients for A and b_{0} is higher than that for Γ for all harmonic degrees ℓ. This shows that the mode amplitude and background power decrease rapidly while the mode width increases slowly with increasing distance from the disc centre. Similar changes are also seen in the corresponding plots for correlation coefficients R_{1} and R_{2} (Fig. 6).
Regression coefficients C_{1} and C_{2} for A increase in magnitude with increasing harmonic degree (ℓ), but the mode width does show a significant relation with ℓ in regression as well as in correlation coefficients. Figure 6 shows a stronger anticorrelation between the quadratic term in the mode parameter A and in λ than that of the linear term. There is a weak positive correlation between the distance from the disc centre and the mode width. But the magnitudes of the correlation are very weak, <0.10. Regression coefficients C_{1} and C_{2} for background power decrease with increasing ℓ; this is also evident from the significant anticorrelation between b_{0} and λ. The correlation coefficients for the background power changes with ℓ and radial order n. Thus we find that the background power is predominantly a function of harmonic degree (ℓ), decreasing with increasing ℓ.
Similarly, the effects of latitude (θ), P angle (P) and B angle (B_{0}) on the mode parameters can be illustrated. The foreshorteningcorrected mode parameters () were obtained by subtracting the first and secondorder terms (see Eq. (10)) from the common modeparameters (). The foreshortening effects on the pmode amplitude and width have also been studied previously (e.g., Howe et al. 2004). Our results support these results.
3.4.2. Effects of duty cycle on the mode parameters
The duty cycle coefficient C_{9} for the mode amplitude A decreases with increasing ℓ (Figs. 5). It is positive for almost all modes at lower degree (ℓ ≲ 600). In the intermediate range of degree, a few modes have negative C_{9}, but with lower magnitudes. The positive value of the coefficient C_{9} shows that the mode amplitude increases with increasing duty cycle, while negative C_{9} correspond to the opposite relation. The increase of A with f is obvious from the increase in signal samples (or number of observed Dopplergrams) in the data cubes, but the reverse is not clear. This may be attributed to the distribution of observed Doppler frames in the data cubes. However, the magnitude of the correlation coefficients C_{9} is very low for degree ℓ > 450.
The regression coefficients C_{9} for the mode width and background power is negative at all degrees (Fig. 5). The magnitude of C_{9} is lowest at lower ℓ and decreases with increasing ℓ. Figure 6 also shows the anticorrelation between the mode width and the duty cycle. But the anticorrelation coefficients decrease with increasing harmonic degree ℓ. The above values of C_{9} and R_{9} show that the mode width and background power decrease with increasing duty cycle.
Linear regression (y = a + b δB) and Pearson correlation coefficients (r) obtained from the fractional mode differences (y) in three frequency bands and magneticactivity indices (δB).
Fig. 8 Frequency averages of the fractional differences in mode amplitude (left two columns) and mode width (right two columns) for flaring (left) and dormant (right) ARs as a function of the magneticactivity index difference (δB = B_{AR} − B_{QR}) of the AR and the corresponding QR. Red and magenta symbols correspond to the ARs flare(s) of magnitude (m) > X3 and X1 ≤ m ≤ X3, respectively. The solid lines show the linear regression fit, while dashed curves show 90% confidence level of the linear fit. Note that the mode parameters are corrected for the foreshortening and duty cycles. (This figure is available in color in electronic form.) 

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Fig. 9 Similar to Fig. 8, but for the background power (left two columns) and mode energy (right two columns). (This figure is available in color in electronic form.) 

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4. Results and discussions
4.1. Variation in pmode parameters with magneticactivity index
To analyse the variation in mode parameters with MAI (B) of flaring and dormant ARs, we applied the mode corrections for foreshortening and duty cycle to all the mode parameters of all the ARs and QRs.
Figure 7 shows the averaged mode parameters in QRs, dormant ARs, and flaring ARs as a function of frequency. Evidently, the average mode amplitudes in QRs are larger than those in dormant and flaring ARs for all radial orders. Flaring ARs, possessing the highest B values, show the smallest average mode amplitude in all radial orders. This is obviously due to the stronger modepower suppression in ARs, which have a stronger MAI (B) than the QRs. The average background power for n ≤ 2 in ARs and QRs varies with B similar to the average mode amplitude. For n ≥ 3, the background power is strongest in flaring ARs and weakest in QRs. The average mode widths are largest in flaring ARs and smallest in QRs for all radial orders, which shows lifetimes of modes in flaring ARs shorter than in dormant ARs and QRs. The mode area (A × Γ) of ARs and QRs shows similar trend as the mode amplitude.
Figures 8 and 9 show the frequency averages of the fractional difference in mode parameters as a function of the MAI difference (δB = B_{AR} − B_{QR}) between ARs and corresponding QRs for flaring and dormant ARs in three frequency bands, as indicated. The coefficients of linear regression and Pearson correlation between the frequencyaveraged fractional mode differences in the three frequency bands and δB are given in Table 2. Averaging the mode parameters in these frequency bands would somewhat hide the frequencydependent properties of the modes.
Previous statistical studies of several ARs showed that mode amplitude and width are linearly related (e.g., Howe et al. 2004; RabelloSoares et al. 2008). Therefore, to find the harmonic degree and frequencydependent relation between mode parameters and δB, we fitted the fractional mode differences (ℛ) and δB with linear regression, ℛ_{n,ℓ}(δB) = α_{0} + α_{1} (δB). The slope (α_{1}) of the linear regression then corresponds to the mode parameter variation per Gauss, hereafter the “parameter variation rate”. The coefficients α_{1} of the linear regression for different mode parameters are shown in Figs. 10 and 11 as a function of harmonic degree (ℓ) and frequency (ν), respectively. The constant term α_{0} has no intrinsic meaning and is not illustrated. In the following, we discuss various mode parameters.
Fig. 10 Coefficients of linear regression in δB (left two columns) and Pearson correlation (right two columns), for different mode parameters of flaring and dormant ARs as a function of harmonic degree. 

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Fig. 11 Similar to Fig. 10, but as a function of frequency. 

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4.1.1. Mode amplitude
Figure 8 (left two columns) shows the frequencyaveraged fractional differences in mode amplitudes (δA/A) in three frequency bands. For both flaring and dormant ARs, δA/A are below zero, that is, negative. However, they are more negative for flaring ARs than for dormant ARs. The amplitude difference decreases with increasing δB, which is further confirmed by the significant correlation coefficients (r) between δA/A and δB (see Table 2). The magnitude of the correlation coefficients is larger in the lower frequency band than in higher frequency bands for flaring ARs. In the flaring ARs, the power suppression per unit Gauss is stronger than in dormant ARs, as is evident from the linear regression slope b (see Table 2). Furthermore, in the fiveminute band of dormant ARs, we infer a stronger suppression than in the lower and higher frequency bands.
The amplitude variations for both flaring and dormant ARs are well correlated with δB, as the evident from the magnitude of Pearson correlation coefficients (see Figs. 10 and 11). This confirms the linear relation between amplitude variation and B (RabelloSoares et al. 2008).
Figures 10 and 11 show changes in the linear regression coefficient α_{1} for the mode amplitude (i.e., rate of change of δA/A per unit Gauss) as a function of harmonic degree (ℓ) and frequency (ν), respectively. For the flaring ARs, the regression coefficients α_{1} increase slowly with harmonic degree ℓ. The correlation coefficients also decrease in magnitude with increasing ℓ. Furthermore, the correlation coefficients for both flaring and dormant ARs change with harmonic degree. For the dormant ARs, the magnitude of the correlation coefficients for different radial orders first decreases with harmonic degree (ℓ), becomes smallest at ℓ in the range ~450–600, and then increases again. This is consistent with the results Bogdan et al. (1993), who reported maximum absorption around wavenumber k ≈ 0.8 Mm^{1} or ℓ ≈ 550. This value of ℓ approximately agrees with our values, where we find strongest suppression of power for n = 1, 2, and 3 modes. For the flaring ARs, the correlation coefficients monotonically increase with degree, except for a few modes at lower degree, for all the radial orders. The significant correlation coefficients further support the relationships between δA/A and δB of the flaring and dormant ARs.
Figure 11 (first row from top) shows that the amplitude change in dormant ARs with frequency is not monotonic. The magnitude of the regression coefficient α_{1} of the dormant ARs decreases with frequency and becomes smallest in the fiveminute band, then decreases. For flaring ARs, α_{1} does not show a significant frequency dependence.
The amplitude decrease rate (i.e., α_{1}) for dormant ARs increases with frequency and peaks around 3.0 mHz and 3.5 mHz for the radial orders n = 1 and n = 2, respectively. This supports the previous reports (Rajaguru et al. 2001; Howe et al. 2004; RabelloSoares et al. 2008). For global modes, Komm et al. (2000a) found about 7% decrease in mode amplitude with the solar cycle. They also found that the strongest variations of 29% occur in the frequency range of 2.7–3.3 mHz. More interestingly, we find that the amplitude variation rate for dormant ARs increases at higher frequencies. The observed increase of the mode absorption as a function of frequency and decrease at higher frequencies have been theoretically reported previously (Jain et al. 2009). Negative values of α_{1} may be attributed to the power absorption (Braun et al. 1987). But it is not clear whether the mode power absorption can be compared with the suppression of power in the sunspot region, because the absorption was calculated for travelling waves, while the modes we analysed are standing waves.
Lower values of the coefficients α_{1} for the flaring ARs than for the dormant ARs may be the combined result of both mode power absorption by sunspots and amplification by flares. However, the absorption effect clearly dominates, as is evident from the negative fractional difference in mode amplitude (Fig. 8).
4.1.2. Mode width
The mode width of a peak profile in the power spectrum is related to the imaginary part of the frequency and hence is a measure of mode damping. It is inversely proportional to the lifetime of the mode. The frequencyaveraged fractional difference in mode width (δΓ/Γ) for flaring and dormant ARs as a function of MAI difference (δB) is shown in Fig. 8 (right two columns). δΓ/Γ of both the flaring and dormant ARs are positive and increase with increasing δB, employing the decrease in lifetime of modes with B. In the fiveminute band of both the flaring and dormant ARs, δΓ/Γ increases faster than in lower and higherfrequency bands. This relation is further confirmed from the high correlation coefficients (see Table 2). But the correlations are better in the fiveminute band of the dormant ARs and in the lowerfrequency band of flaring ARs. The higher values of the intercept a > 0 for dormant ARs than flaring ARs show that the mode width is larger in ARs than in QRs. Thus it appears that most modes live longer in QRs than they do in ARs, employing the higher damping in ARs. For dormant ARs, data points are closer to the linear regression line, while for flaring ARs there is a large spread at larger δB, causing lower correlations. This may be caused by activities in the flaring ARs. Furthermore, the steeper slope in the fiveminute band of flaring ARs than in dormant ARs shows that the mode width in flaring ARs decreases more slowly with δB than in dormant ARs.
The coefficients (α_{1}) of the linear regression between the fractional difference in mode width and δB are shown in Figs. 10 and 11 (second row from the top) as a function of harmonic degree (ℓ) and frequency (ν), respectively. The coefficients α_{1} for both dormant and flaring ARs are positive and increase with ℓ. Its values are lower for flaring ARs than the dormant ARs for all ℓ except for a few modes with ℓ < 350. The highest value of the width variation for flaring and dormant ARs occurs in the range ~1–2 Gauss^{1} and ~2–6 Gauss^{1}, respectively.
Chen et al. (1996), using the absorption of pmode waves in sunspots as a tool to determine the lifetime of pmodes in the ℓ range (200–700), reported that pmode lifetimes decrease with ℓ and frequency (ν). The widthvariation rate of dormant ARs initially increases with frequency for all radial orders, peaks at different frequencies in the fiveminute frequency band, and then decreases (except n = 0). For radial order, n = 0, the maximum of about 0.0032 Gauss^{1} occurs around ν ≈ 2.6 mHz. But note that we did not cover the modes ν > 2.6 for n = 0. Rajaguru et al. (2001), Howe et al. (2004), and RabelloSoares et al. (2008) have reported the maximum around 0.005, 0.0045 and 0.003 Gauss^{1}, respectively.
For the global modes, Komm et al. (2000a) reported solar cyclic modewidth variations of about 3% and the maximum changes of 47% in the frequency range 2.7–3.5 mHz. But they did not find ℓ dependence of the solar cycle changes in mode width, presumably because of weaker variations in the average global magnetic fields.
4.1.3. Background power
The background power in solar oscillation spectra is not just “noise”, but contains physical information that might be important for a better understanding of the dynamics of the solar atmosphere. It has a large component of socalled solar noise, which is the background produced by convective cells. Furthermore, the detection of oscillation modes essentially depends upon the signaltonoise ratio, therefore, it is important to compare the background noise in the power spectra between the ARs and corresponding QRs. In addition, it should be noted that the mode background power may contain substantial contributions from mode “tails”, particularly when the mode is asymmetric because it is not taken into account in the fitting (cf., Eq. (4)).
Figure 9 (left two columns) shows that the relative variation in frequencyaveraged background power of dormant ARs decreases slowly with increasing δB while it is almost constant for flaring ARs. For most of the ARs, the background power is weaker than their corresponding QRs, which confirms previous reports (Rajaguru et al. 2001). The reduction is expected because the magnetic field is known to suppress convection, which is the main source of background in power spectra. But the fractional difference in background power at lowerfrequency band is larger than zero for a few modes with radial order, for example, n = 2 (see Fig. 9). The correlation between δb_{0}/b_{0} and δB is poor or nonexistent in flaring ARs, while there is significant anticorrelation for dormant ARs (see Table 2). The poor correlation coefficient for flaring ARs may be caused by mode suppression due to sunspots and amplification due to flares.
Komm et al. (2000a) found no significant variation in background power with solar cycle in the global modes. The expected variation in background power with solar cycle in their analysis may be due to the weak increase in the average magnetic field with the solar cycle. Moreover, the global mode analysis is restricted to the ℓ < 200, where the variation is weak even in our results.
The coefficients of linear regression (α_{1}) between the fractional difference in background power and δB are shown in Figs. 10 and 11 (third row from the top) as a function of ℓ and ν, respectively. The coefficient α_{1} is negative (positive) for most flaring (dormant) ARs with n < 2. For pmodes of flaring ARs, it first decreases with frequency and becomes lowest at different frequencies in the fiveminute band for different radial orders, and then increases. The negative value of the coefficients α_{1} furthermore shows the opposite relation between background power and MAI, B. The positive value of α_{1} for flaring ARs shows that power enhancement may be caused by flare induced excitation. However, the correlation coefficients for most of the modes are very low for flaring ARs.
Fig. 12 Variations in the rate of fractional differences in mode amplitude and mode width of flaring (left) and dormant (right) ARs. Solid lines show the linear regression fit with the fitted coefficients as given in the respective equations. (This figure is available in color in electronic form.) 

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4.1.4. Mode energy
If the total power in the mode is the area under the peak in the power spectrum, it is not instantly clear whether the increase in line widths in ARs compensates for the decrease in peak height. The area below a peak is a measure of excitation for the corresponding mode. To examine these changes, we plotted the fractional difference of the product of amplitude and width as a proxy for the area below the peak or mode energy (see Eq. (7)) in three frequency bands. This is shown in Fig. 9 (right two columns).
Figure 9 shows that the mode energy for both flaring and dormant ARs is lower than in the corresponding QRs and decreases with increasing δB. But the decrease rate in dormant ARs is faster than in flaring ARs, as is evident from the linear regression slope (see Table 2). In the dormant ARs the energy difference decreases faster in the fiveminute band than in the lower and higherfrequency bands. The intercept (a), and also the anticorrelation coefficient decrease from lower to higher frequencies. For the global modes, Komm et al. (2000a) reported a reduction of up to 60% in mode area. They found that the highest reduction of 36% in mode area occurs in the frequency range of 2.7–3.3 mHz.
The linear regression fit shows that the slopes of the flaring ARs are very small for the fiveminute band. This implies that the mode energy slowly decreases with B of flaring ARs. This is perplexing because several studies have reported strong pmode power absorption by AR magnetic fields (Braun et al. 1987; Braun & Duvall 1990; Rajaguru et al. 2001; Mathew 2008). Therefore, the mode energy is expected to decrease faster with increasing B instead of the rate as, is inferred by the small slope for the energy difference for the flaring ARs in Fig. 9.
Figures 10 and 11 (bottom panel) show the coefficients of linear regression α_{1} between the fractional mode energy difference δE/E and δB as a function of harmonic degree and frequency, respectively. The energy variation rate (i.e., α_{1}) for dormant ARs initially decreases with increasing degree ℓ, and becomes lowest around ℓ ≈ 500 for all the radial orders then increases. We note that there is a contribution to the mode energy from the instrumental resolution, which is effectively around 5 arcsec for GONG due to seeing. Previous studies have shown a decrease in mode energy with harmonic degree at a fixed frequency (e.g. Rhodes et al. 1991). Woodard et al. (2001) have also studied the energy rate in the intermediate mode. They inferred that the timeaveraged energy per mode, which is theoretically related to the modal surface velocity power, decreases steeply with ℓ at a fixed frequency over the entire observed ℓrange. Specifically, at ν = 3.1 mHz, the energy per mode drops by a factor of ~10 between ℓ = 150 and ℓ = 650. For flaring ARs, the mode energy variation rate is significantly different from the dormant ARs. α_{1} for flaring ARs increases with increasing ℓ except for few modes. On average, α_{1} of flaring ARs is smaller in magnitude than in dormant ARs, indicating lower mode energy in flaring ARs than in dormant ARs.
The abovementioned difficulty may be resolved as follows: Note that we computed the mode energy using data cubes corresponding to the ARs of our sample during their maximum flaring periods. Hence, the mode energy would include the net result of absorption due to the sunspots and amplification due to the flares. In ARs with a strong magnetic fields, that is, high magnetic index (B), but lower flareactivity, that is, low flare index (FI), the effect due to mode absorption would dominate. On the other hand, when the flare energy is higher than the energy absorbed by sunspots, the amplification effect due to flares would dominate. We suggest that the small slope in pmode energy (Fig. 9), which does not show a significant decrease with increasing B, could be attributed to the increasingly stronger flaring activity in the magnetically complex, flareproductive ARs. However, it is not clear whether the variations in A or in Γ are contributing more to the changes in mode energy. In the following, we attempt to analyse this problem in more detail.
Fig. 13 Frequency averages of the fractional difference in mode amplitude, background power, mode width, and mode energy of flaring ARs, from top to bottom, as a function of flareactivity index. Mode parameters are corrected for foreshortening, duty cycle and magneticactivity. Red and magenta symbols correspond to ARs flare(s) of magnitude (m) > X3 and X1 ≤ m ≤ X3, respectively. The solid lines show the linear regression, and the dashed curves correspond to the 95% confidence level of the linear fit. (This figure is available in color in electronic form.) 

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To study the major contribution to the mode energy variation, we plot the coefficients of the fractional amplitude variation and width variation for flaring and dormant ARs (see Fig. 12). For dormant ARs, we find that the fractional increase in mode width is followed by the fractional decrease in mode amplitude, supporting previous reports (RabelloSoares et al. 2008). However, the increase in width is slightly faster than the decrease in amplitude, as is evident from the negative slope of the linear regression. This shows that the fractional decrease in mode energy of dormant ARs is caused by the decrease in amplitude as well as in width, but the width increase contributes slightly more than the decrease in amplitude.
For the flaring ARs, the trend is quiet different from that of the dormant ARs. The increase in fractional difference in mode width is followed by the increase in fractional difference in mode amplitude and the resulting increase in fractional mode energy. This can be seen in Figs. 10 and 11 (bottom panel). The fractional increase in mode width is somewhat stronger than the fractional increase in mode amplitude. This shows that the net contribution to the variation in fractional mode energy is dominated by the mode width.
4.2. Changes in mode parameters with flaring activity
From the previous Sect. 4.1, we learned that the pmode properties of flaring ARs are distinctly different from those of the dormant ARs. We have weaker mode suppression and lower width variation rates in flaring ARs than in dormant ARs. To study the flarerelated changes in mode parameters, we corrected the mode parameters, of all the flaring ARs and corresponding QRs for foreshortening, duty cycle, and magneticactivity. From Sect. 4.1 we found that the mode amplitude and width show a linear relation with B in agreement with the reports by RabelloSoares et al. (2008). We fitted the foreshortening and dutycyclecorrected mode parameters of all the ARs and QRs as a function of B, . Then every mode parameter of flaring ARs and corresponding QRs were corrected for the MAI using the coefficients ξ obtained from the fitting according to , where B_{min} is the lowest value of B in the data sample. We repeated this analysis for every common multiplet (n,ℓ) for mode parameters A, b_{0}, and Γ. Then we computed the fractional difference in mode parameters between flaring ARs and corresponding QRs (e.g., for mode energy see Eq. (7)).
Fig. 14 Coefficients of linear regression in FI (top row) and Pearson correlation (bottom row) for different mode parameters of flaring ARs as a function of harmonic degree. 

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Figure 13 shows the frequency average of the fractional mode difference of corrected modeparameters as a function of the flareactivity index (FI). We computed the linear regression and correlation coefficients between the frequency averages of the fractional differences of mode parameters and flare indices, which are listed in Table 3. Figures 14 and 15 show the coefficients of linear regression and Pearson correlations between fractional difference in mode parameters and FI as a function of harmonic degree (ℓ) and frequency (ν), respectively. In the following, we analyse and discuss the various mode parameters.
Linear regression (Y = α + β FI) and Pearson correlation coefficients (R) obtained from the frequencyaveraged fractional mode difference (Y) in three frequency bands and flareactivity index (FI) of the flaring ARs.
4.2.1. Mode amplitude
Figure 13 (top panel) shows that the mode amplitude of flaring ARs increases with FI in all frequency bands. In the fiveminute band the increase in δA/A is stronger than the lower and higherfrequency bands, as is evident from the linear regression slope for the fiveminute band (see Table 3). Moreover, the correlation between δA/A and FI is higher for the fiveminute and higherfrequency bands than in the lowerfrequency band. The positive steep slope shows flareassociated enhancement in mode amplitude. Our results of flare induced amplification in mode amplitude supports previous reports (Ambastha et al. 2003; Maurya et al. 2009).
However, several data points with Log(FI) ≤ 9 are seen below δA/A = 0, showing a smaller amplitude in flaring ARs than in the corresponding QRs. Similar results have also been reported by Ambastha et al. (2003) for some flaring ARs. Figures 14 and 15 (left column) show the linear regression coefficient (β_{1}) as a function of harmonic degree (ℓ) and frequency (ν), respectively. The amplitude variation rate increases with ℓ for all the radial orders. It increases with frequency and becomes highest in the fiveminute band, then decreases. We also find a significant correlation for degree ℓ > 300 and ν > 2500.
Fig. 15 Similar to Fig. 14, but as a function of frequency. 

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4.2.2. Mode width
Figure 13 (third row from the top) shows the frequency average of the fractional difference (δΓ/Γ) in mode width as a function of FI of the flaring ARs. δΓ/Γ in the fiveminute and higherfrequency bands decreases with increasing FI, while it shows the opposite trend in the lowerfrequency band. The relation between fractional difference in mode width and FI is also evident from the regression slope and the correlation coefficients given in Table 3. There is a strong correlation between relative width and FI for the fiveminute and higherfrequency band, but the correlation is poor at the lower frequency band. Ambastha et al. (2003) have found a decrease in mode width during flares in some ARs, but no such signatures for some other flaring ARs in their data samples. Tripathy et al. (2008) have found that the CMEprone ARs having lower values of magnetic flux have a smaller line width than the QRs.
Figures 14 and 15 (second column from the left) show the linear regression coefficients β_{1} for relative mode width as a function of harmonic degree and frequency, respectively. The coefficient β_{1} is very low for harmonic degree ℓ < 450 and decreases with increasing ℓ(>450). The correlation coefficients (r) also show a similar trend as the parameter β_{1}. The coefficient β_{1} decreases with frequency for different radial orders (Fig. 15). But for the modes with radial order n = 1, 2, β_{1} is smallest in the fiveminute band at different frequencies, then it increases. The decrease in mode width with FI indicates an increase in the mode lifetimes.
4.2.3. Background power
Figure 13 (second row from the top) shows that the fractional difference (δb_{0}/b_{0}) in background power increases with FI in all three frequency bands; the slope steepens with increasing frequency. The slope is steeper in the fiveminute band than in lower and higherfrequency bands, similar to the amplitude. The correlation coefficient is also higher in the fiveminute frequency band than in the lower and higherfrequency bands as is evident from Table 3. The positive slope at all frequency bands of the background power furthermore suggests the flareinduced amplification in mode power. But the correlation in the background power and FI are not significant. The background power variation with degree (Fig. 10) and frequency (Fig. 11) show similar relations.
Fig. 16 Variations in the rate of fractional difference in mode amplitude and width of flaring ARs. The solid line shows the secondorder polynomial fit; the fitted coefficients are given in the equation. (This figure is available in color in electronic form.) 

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4.2.4. Mode energy
The frequencyaveraged fractional difference in mode energy (δE/E) in all three frequency bands increases with increasing flareactivity index, see Fig. 13 (bottom row). The rate of increase in δE/E for the fiveminute frequency band is highest as is evident from the magnitude of the linear regression slope, which is smaller in the lower and higherfrequency bands (see Table 3). This indicates that flare induced excitation is more significant in the fiveminute bands than in the lower and higher frequencies. The significant positive correlation coefficients between mode energy and FI additionally support these results.
Figures 14 and 15 show that the mode energy variation rate with FI increases with increasing frequency and harmonic degree and peaks in the degree range 400–600 and in the fiveminute frequency band for all radial orders, then decreases at higher degree and frequencies, respectively. To ascertain the contribution of amplitude and width to the energy variation, we plot the width variation rate vs. amplitude variation rate in Fig. 16.
Figure 16 shows that mode amplitude (mode width) variation rate is positive (negative) for most of the modes, and an increase in mode width is followed by a rapid decrease in mode amplitude. This shows that the contribution of the mode amplitude to the variation of the mode energy is stronger than that of the mode width. In the fiveminute frequency band, there is a high positive amplitude variation rate and a relatively low negative mode width rate. However, for a few modes in the lower and higherfrequency bands the positive amplitude variation rate decreases very rapidly, and a positive mode width variation rate occurs.
5. Summary and conclusions
We studied the highdegree pmode properties of a sample of several flaring and dormant ARs and associated QRs, observed during solar cycles 23 and 24 using the ringdiagram technique, assuming plane waves, and their association with magnetic and flare activities. The changes in pmode parameters are the combined effects of duty cycles, foreshortening, magnetic and flare activities, and measurement uncertainties.
The pmode amplitude (A) and background power (b_{0}) of ARs were found to be decreasing with their angular distances from the disc centre, while the width increases slowly. The effects of foreshortening on the mode amplitude and width are consistent with reports by Howe et al. (2004). The decrease in mode amplitude A with distance arises because with increasing distance from the disc centre we measure only the cosine component of the vertical displacement. Moreover, foreshortening causes a decrease in spatial resolution of the Dopplergrams as we observe increasingly closer toward the limb. This reduces the spatial resolution determined on the Sun in the centretolimb direction, and hence leads to systematic observational errors.
The secondlargest effects on pmode parameters are caused by duty cycle. We found that the mode amplitude increases with increasing duty cycle, while the mode width and background power show the opposite trend. Similar results were reported previously for the global pmode amplitude and width, for example by Komm et al. (2000a). These authors reported the strongest increase in mode width and reduction in amplitude with duty cycle when its values are lower. These changes in mode parameters may be caused by the increase in signal samples in data cubes. However, we found that for a few modes in the fiveminute and in higherfrequency bands, the mode amplitudes do not increase significantly with duty cycle. The effect of the duty cycle decreases with increasing harmonic degree ℓ. To study the relation of mode parameters with magnetic and flare activities, we corrected the mode parameters of all the ARs and QRs for foreshortening.
We found that the mode amplitude in ARs is considerably smaller than in QRs. In the dormant ARs, the mode amplitude decreases with increasing B. There is a stronger reduction in the fiveminute band than in the lower and higherfrequency bands. The reduction in mode amplitude of ARs has been reported previously by several researchers (Braun et al. 1987; Braun & Duvall 1990; Rajaguru et al. 2001; Mathew 2008). Goode & Strous (1996) have reported the suppression of acoustic flux and pmode power even in a weak magnetic field QR. However, a precise mechanism of the energy absorption is not yet established. The possible mechanisms for mode power reduction in ARs are (i) Absorption of pmodes within sunspots (Braun et al. 1988). It is assumed that the sunspot magnetic fields play a crucial role in transforming some part of the acoustic energy to pure magnetic energy, for instance, Alfvén waves type (Crouch & Cally 2003, 2005; Cally et al. 2003), which may be transported to the upper atmosphere of the Sun (Marsh & Walsh 2006, and references therein). (ii) The efficiency of the pmode excitation by turbulent convection (Goldreich & Keeley 1977; Goldreich & Kumar 1988, 1990) might be reduced in a magnetic field owing to the nature of magnetoconvection (Hughes & Proctor 1988). (iii) Wilson depression in a sunspot at the height of the formation of spectral lines, which is used to measure the Doppler shift, causes additional phase shifts to the velocity measurements, as compared to nonmagnetic regions. (iv) Modification of the surface values of the pmode eigenfunctions by the magnetic fields of ARs (Hindman et al. 1997). (v) Resonant absorption of solar pmodes by sunspots (Hollweg 1988). (vi) Mode mixing in sunspots (D’Silva 1994). The energy in an incoming mode, at any horizontal wavenumber, is dispersed into a wide range of wave numbers. (vii) Excitation of tube waves through pmode buffeting (Bogdan et al. 1996; Hindman & Jain 2008). (viii) Inhomogeneityenhanced thermal damping (Riutova & Persson 1984). Jain et al. (1996) showed that the horizontal magnetic field can lower the upper turning point and change the skin depth for a simple planeparallel adiabatically stratified polytrope. In addition to power suppression, they also found that magnetic field alters the phase of pmodes.
The inclination of field lines from the vertical can affect the amount of acoustic power absorption (Cally et al. 2003) by conversion of acoustic to slow magnetoacoustic waves. The observational confirmation of the field inclinationrelated variations in mode power have been reported previously from the timedistance (Zhao & Kosovichev 2006) and acousticholography (Schunker et al. 2005, 2008) analyses. In the ringdiagram analysis, we found these effects averaged over larger area.
The mode width in ARs are generally smaller than in the corresponding QRs and increases with magnetic field. This may be caused by strong damping of pmodes in the strong magneticfield areas. Moreover, the width increases with frequency and becomes largest in the fiveminute bands then decreases. But the relation in flaring ARs is poorer than in dormant ARs. This may be caused by flareinduced changes in mode parameters. For stochastically excited modes, a broadening in mode width shows a reduced lifetime or increased damping of the modes in regions of high magneticactivity indices.
A possible mechanism by which magneticactivity can influence mode widths is excitation of oscillations in flux tubes, as suggested by Bogdan et al. (1996) and Hasan (1997). These authors suggested that the flux tubes lead to a balance between energy input from pmodes and losses through radiative damping and leakage from fluxtube boundaries. The excitation of resonant oscillation in flux tubes in unstratified atmosphere was studied by Chitre & Davila (1991) and Ryutova & Priest (1993a,b). Resonant coupling with MHD waves (Pintér & Goossens 1999) might also contribute to the damping of pmodes, as well as scattering of pmodes by the flux tube (Keppens et al. 1994; Bogdan & Zweibel 1987). Thus, when B increases, pmodes are increasingly damped by the interactions with the increasing number of flux tubes. Gascoyne & Jain (2009) have shown that the suppression of sound speed and pressure within the fluxtube region is not the only factor one needs to consider in the scattering of pmodes. There is a direct effect of the magnetic fields caused by the flaring of field lines on the phase shifts.
The combined effects of the mode amplification and width variations appear in mode energy variations. We found that the average mode energy in flaring ARs is lower at all frequencies than in dormant ARs and QRs. The decrease rate of the mode energy in dormant ARs is highest around ℓ = 450, while the highest decrease rate for flaring ARs shifted towards lower ℓ. More interestingly, we found that the increase in width variation rate of dormant (flaring) ARs is followed by decrease (increase) in mode amplitude variation rate. The first one for the dormant ARs confirms previous reports by RabelloSoares et al. (2008). The second one for the flaring ARs is attributed to the flareassociated changes in the oscillations characteristics of pmodes.
The decrease in mode energy implies a reduction in the amount of acoustic energy pumped into the modes by turbulent convection. This agrees with the assumption that the presence of strong magnetic fields suppresses motion in a turbulent medium, which is known to occur for several solar surface activities. For turbulent excitation and damping, the energy of a single oscillation mode is expected to depend mainly on mode frequency (at least for modes whose angular wavenumber is well below that of the excitation turbulence). The implication of energy depending only on frequency is that the surface velocity power of a single oscillation mode should increase with angular degree at fixed ν. Theoretically, Jain et al. (2009) have shown that mode absorption increases rapidly with frequencyreaching a maximum or saturation value near 4 mHz, and decreases at higher frequencies ν > 4 mHz. They investigated the pmode absorption by fibril magnetic field. They suggested that the pmodes excite tube waves through mechanical buffeting on the magnetic fibrils in the form of longitudinal sausage waves and transverse kink waves. The tube waves propagate up and down the magnetic fibrils and out of the pmode cavity, thereby removing energy from the incident acoustic waves.
The background power of the dormant ARs were found to be weaker than in corresponding QRs for radial order n ≤ 2. But for the flaring ARs, the background power variations seem to be a function of harmonic degree and frequency (ν). It decreases with increasing ν and becomes lowest at a specific frequency in the fiveminute frequency band then increases. These changes were attributed to the flareassociated charges in the oscillation modes. The background power of flaring and dormant ARs were found to decrease with MAI, which reinforces the idea that strong magnetic fields could hinder convection (Biermann 1941; Chandrasekhar 1961), which is the source of solar noise.
Magneticfieldinduced activities in the surface and higher layers may play a role in the excitation of oscillation modes in the ARs. To study the effects of flares on pmode parameters, we employed mode corrections for foreshortening, duty cycle, and magneticactivity. We found that the mode parameters show a significant correlation with flareactivity index. The pmode amplitude increases with flareactivity index and shows stronger amplification in the fiveminute band. The increase in background power with flareactivity index furthermore supports the flareassociated excitation in pmodes (Wolff 1972). More interestingly, our statistical study showed an association of pmode energy with flares, supporting the expected mode excitation by flares. The mode width is found to decrease with flareactivity index, indicating an increase in the lifetime of modes or mode damping. The combined effects of mode amplitude and width appeared in the mode energy. The mode energy increases with increasing flareactivity index and more strongly so in the fiveminute band. Our study showed that the increase in the mode energy is mostly contributed by the increase in mode amplitude and not by the increase in width or damping.
This suggests that the energetic solar flares may indeed produce effects in the solar interior below ARs, although the major process of flareenergy release occurs in the external solar atmosphere. However, the exact mechanism of the energy transfer towards the solar photosphere and flare induced excitation in mode power is not yet known. The detailed mechanism of the energy transfer from the corona to the photosphere is not well understood. Wolff (1972) was probably the first to propose the mechanism of momentum transfer toward the photosphere from the flareenergy release site to describe the possible impulse for the pmode excitation. Recently, Hudson et al. (2012) proposed a similar mechanism of momentum transfer to explain flareinduced excitation of the seismic waves. They listed four basic mechanisms proposed previously: (i) a hydrodynamic shockwave originating in the chromosphere (Kostiuk & Pikelner 1975; Kosovichev & Zharkova 1998); (ii) Lorentz force from the magnetic transients (Anwar et al. 1993; Kosovichev & Zharkova 2001; Sudol & Harvey 2005; Hudson et al. 2008); (iii) photospheric backwarming (Machado et al. 1989; MartínezOliveros et al. 2008); and (iv) “McClymont magnetic jerk” (Hudson et al. 2008). Work done by Lorentz forces in the backreaction could supply enough energy to explain observations of flaredriven seismic waves (Hudson et al. 2008). The requirement for momentum conservation can in principle help to distinguish among these plausible mechanisms (Hudson et al. 2012). Hudson et al. (2008) introduced the idea of the coupling of flare energy into a seismic wave, namely the “McClymont magnetic jerk”, produced during the impulsive phase of acoustically active flares.
Maurya & Ambastha (2009) and Maurya et al. (2012) have reported large Xclass whitelight flares in seismically active ARs. Pedram & Matthews (2012) have also studied the hard Xray characteristics of seismically active and quiet whitelight flares. They found that the acoustically active flares are associated with a larger and more impulsive deposition of electron energy. However, they argued that this does not always correspond to a higher whitelight contrast. Unfortunately, observations of whitelight flares are rare. We suggest that the Xray flux alone does not seem to be an adequate measure of how much energy a flare deposits in the photosphere. This is mainly because of the extent of soft Xray emission produced at higher coronal heights. A lower Xclass but larger Hαclass or a whitelight flare may be more important for affecting the solar acoustic oscillation modes.
We found that the flare index (FI), as calculated in this study, is a poor indicator for the helioseismic effects of a flare on the photosphere and on the oscillation characteristics because it is primarily based on the Xray flux alone. Moreover, a higher FI value may result from several relatively longerduration flares of smaller magnitude integrated over the time internal of the ring data cube. These smallmagnitude flares that contribute to the larger FI may not be able to individually affect the oscillation characteristics. On the other hand, it is likely that a shortduration, energetic, and impulsive flare of large magnitude is seismically more effective, but may give rise to relatively smaller FI.
The MAI (MAI or B) used in this analysis was obtained from the lineofsight components of the full magnetic field strength. The mode power absorption also depends upon the inclination of field lines from the vertical, as stated above. This may give rise to some systematic errors in the analysis of the relationship of mode parameters and magneticactivity. Different magneticfield configurations in ARs may give different systematic errors. It is difficult to account for the order of the systematic errors, but continuous observations of vector magnetic fields can be used to analyse the errors.
This study supports previous results by Ambastha et al. (2003) and Maurya et al. (2009) that large flares are able to significantly amplify the highdegree pmodes, over and above the mode absorption by strong magnetic fields of ARs. The associated pmode parameters are also affected by the flareinduced changes, thereby affecting the subsurface properties that are derived using the computed pmodes. Therefore, we suggest that adequate care should be taken in describing the subsurface properties of ARs especially while using the photospheric pmodes. We plan to analyse in depth the frequency shift, subphotospheric flow, sound speed, etc. in these active regions in our future studies.
Acknowledgments
This work used data obtained by the GONG program operated by AURA, Inc. and managed by the National Solar Observatory under a cooperative agreement with the National Science Foundation, USA The ringdiagram analysis performed using the GONG pipeline. The integrated Xray flux data were obtained from GOES, which is operated by the National Oceanic and Atmospheric Administration, USA The solar activity information provided by the solar monitors web pages helped us to select the flaring and dormant active regions. The authors would like to thank H.M. Antia and F. Hill for their useful discussion and suggestions. Thanks are also due to the anonymous referee for his comments which helped in improving the paper. R.A.M. and J.C. acknowledge support by the National Research Foundation of Korea (20110028102 and NRF2012R1A2A1A03670387).
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All Tables
Linear regression (y = a + b δB) and Pearson correlation coefficients (r) obtained from the fractional mode differences (y) in three frequency bands and magneticactivity indices (δB).
Linear regression (Y = α + β FI) and Pearson correlation coefficients (R) obtained from the frequencyaveraged fractional mode difference (Y) in three frequency bands and flareactivity index (FI) of the flaring ARs.
All Figures
Fig. 1 Distribution of flaring (filled circle) and dormant (unfilled circle) ARs used in this study as shown in a Carrington map. Circle sizes correspond to the magneticactivity index (B; for detail see Sect. 3.3) of ARs in Gauss. 

Open with DEXTER  
In the text 
Fig. 2 pmode parameters obtained for AR NOAA 10649 on 17 July 2004: a) harmonic degree ℓ; b) mode amplitude A; c) mode width Γ; d) background power b_{0}; and e) mode area A × Γ as a function of frequency for radial orders n = 0,...,5. 

Open with DEXTER  
In the text 
Fig. 3 Relation between magneticactivity index (B) and flareactivity index (FI) of flaring ARs. Filled circles represent flaring ARs of our sample. The solid line shows the linear regression line, while the dashed lines around it correspond to the 95% confidence levels of the linear fit. 

Open with DEXTER  
In the text 
Fig. 4 Mode parameters, amplitude A (top row), background power b_{0} (middle row), and width Γ (bottom row) for two modes (n = 0,ℓ = 652 (left three columns); n = 1,ℓ = 449 (right three columns)) for dormant ARs and QRs as a function of the central meridian distance λ of the region centres (left), duty cycle f (middle), and magneticactivity index B (right). Dashed curves show the quadratic fit (in λ) and the linear fit (in f and B) through the mode parameters. 

Open with DEXTER  
In the text 
Fig. 5 Multiple nonlinear regression coefficients, C_{1} (top row), C_{2} (middle row), and C_{9} (bottom row), normalized with intercept, C_{0}, for the mode parameters, A (left column), Γ (middle column), and b_{0} (right column) as a function of harmonic degree, ℓ. Corresponding correlation coefficients are shown in Fig. 6. 

Open with DEXTER  
In the text 
Fig. 6 Correlation coefficients, R_{1} (top row), R_{2} (middle row), and R_{9} (bottom row) corresponding to first and secondorder terms in central meridian distance λ (see Eq. (10)), respectively, for the mode parameters, A (left column), Γ (middle column), and b_{0} (right column) as a function of harmonic degree, ℓ. 

Open with DEXTER  
In the text 
Fig. 7 Average common modeparameters in QRs (black), dormant ARs (DARs; blue), and flaring ARs (FARs; red) for different radial orders as a function of mode frequency. Note that mode parameters are corrected for foreshortening and duty cycle before averaging. (This figure is available in color in electronic form.) 

Open with DEXTER  
In the text 
Fig. 8 Frequency averages of the fractional differences in mode amplitude (left two columns) and mode width (right two columns) for flaring (left) and dormant (right) ARs as a function of the magneticactivity index difference (δB = B_{AR} − B_{QR}) of the AR and the corresponding QR. Red and magenta symbols correspond to the ARs flare(s) of magnitude (m) > X3 and X1 ≤ m ≤ X3, respectively. The solid lines show the linear regression fit, while dashed curves show 90% confidence level of the linear fit. Note that the mode parameters are corrected for the foreshortening and duty cycles. (This figure is available in color in electronic form.) 

Open with DEXTER  
In the text 
Fig. 9 Similar to Fig. 8, but for the background power (left two columns) and mode energy (right two columns). (This figure is available in color in electronic form.) 

Open with DEXTER  
In the text 
Fig. 10 Coefficients of linear regression in δB (left two columns) and Pearson correlation (right two columns), for different mode parameters of flaring and dormant ARs as a function of harmonic degree. 

Open with DEXTER  
In the text 
Fig. 11 Similar to Fig. 10, but as a function of frequency. 

Open with DEXTER  
In the text 
Fig. 12 Variations in the rate of fractional differences in mode amplitude and mode width of flaring (left) and dormant (right) ARs. Solid lines show the linear regression fit with the fitted coefficients as given in the respective equations. (This figure is available in color in electronic form.) 

Open with DEXTER  
In the text 
Fig. 13 Frequency averages of the fractional difference in mode amplitude, background power, mode width, and mode energy of flaring ARs, from top to bottom, as a function of flareactivity index. Mode parameters are corrected for foreshortening, duty cycle and magneticactivity. Red and magenta symbols correspond to ARs flare(s) of magnitude (m) > X3 and X1 ≤ m ≤ X3, respectively. The solid lines show the linear regression, and the dashed curves correspond to the 95% confidence level of the linear fit. (This figure is available in color in electronic form.) 

Open with DEXTER  
In the text 
Fig. 14 Coefficients of linear regression in FI (top row) and Pearson correlation (bottom row) for different mode parameters of flaring ARs as a function of harmonic degree. 

Open with DEXTER  
In the text 
Fig. 15 Similar to Fig. 14, but as a function of frequency. 

Open with DEXTER  
In the text 
Fig. 16 Variations in the rate of fractional difference in mode amplitude and width of flaring ARs. The solid line shows the secondorder polynomial fit; the fitted coefficients are given in the equation. (This figure is available in color in electronic form.) 

Open with DEXTER  
In the text 