Surface-Focused Seismic Holography of Sunspots: I. Observations

We present a comprehensive set of observations of the interaction of p-mode oscillations with sunspots using surface-focused seismic holography. Maps of travel-time shifts, relative to quiet-Sun travel times, are shown for incoming and outgoing p modes as well as their mean and difference. We compare results using phase-speed filters with results obtained with filters that isolate single p-mode ridges, and we further divide the data into multiple temporal frequency bandpasses. The f mode is removed from the data. The variations of the resulting travel-time shifts with magnetic-field strength and with the filter parameters are explored. We find that spatial averages of these shifts within sunspot umbrae, penumbrae, and surrounding plage often show strong frequency variations at fixed phase speed. In addition, we find that positive values of the mean and difference travel-time shifts appear exclusively in waves observed with phase-speed filters that are dominated by power in the low-frequency wing of the p ₁ ridge. We assess the ratio of incoming to outgoing p-mode power using the ridge filters and compare surface-focused holography measurements with the results of earlier published p-mode scattering measurements using Fourier?–?Hankel decomposition.     

Analysis of the Helioseismic Power-Spectrum Diagram of a Sunspot

The continuous high spatial resolution Doppler observation of the Sun by the Solar Dynamics Observatory/Helioseismic and Magnetic Imager allows us to compute a helioseismic k–ω power-spectrum diagram using only oscillations inside a sunspot. Individual modal ridges can be clearly seen with reduced power in the k–ω diagram that is constructed from a 40-hour observation of a stable and round sunspot. Comparing this with the k–ω diagram obtained from a quiet-Sun region, one sees that inside the sunspot the f-mode ridge is more reduced in power than the p-mode ridges, especially at high wavenumbers. The p-mode ridges all shift toward lower wavenumber (or higher frequency) for a given frequency (or wavenumber), implying an increase of phase velocity beneath the sunspot. This is probably because the acoustic waves travel across the inclined magnetic field of the sunspot penumbra. Line-profile asymmetries exhibited in the p-mode ridges are more significant in the sunspot than in the quiet Sun. Convection inside the sunspot is also highly suppressed, and its characteristic spatial scale is substantially larger than the typical convection scale of the quiet Sun. These observational facts demand a better understanding of magnetoconvection and interactions of helioseismic waves with magnetic field.     

Variations in Oscillation Frequencies From Minimum to Maximum of Solar Activity

The temporal variation in intermediate-degree-mode frequencies is analysed using helioseismic data which cover the minimum to the maximum phase of the current solar cycle. To study the variation in detail, the measured frequency shifts of f and p modes are decomposed into two components, viz., oscillatory and non-oscillatory. The f-mode frequencies exhibit prominent oscillatory behavior in contrast to p modes where the oscillatory nature of the frequencies is not clearly seen. Also, the oscillatory part contributes significantly to the f-mode frequencies while p-mode frequencies have maximum contribution from the non-oscillatory part. The amplitude of both oscillatory and non-oscillatory parts is found to be a function of frequency. The non-oscillatory part is observed to have a strong correlation with solar activity.     

Analytical signal as a tool for studying solar p-modes

A technique for analyzing periodic processes based on the introduction of an analytical signal is described. This technique allows the instantaneous frequency, amplitude, and phase of oscillations to be obtained. The data on solar brightness fluctuations collected with the DIFOS multichannel photometer onboard the CORONAS-F satellite are processed. The p-mode spectral lines are broadened mainly by amplitude fluctuations, while the frequency stability appears to be high (～10^?4). A method for separating signals with close frequencies is developed. The p-mode with l = 0 and n = 21 is used as an example to show that the separation of signals with close frequencies is possible when the conventional spectral methods are inefficient. Analysis of the phase shifts between the oscillations observed in various optical channels of the DIFOS photometer has revealed that the five-minute oscillations travel from the upper and deep photospheric layers toward the middle photospheric layers. This effect directly proves that the evanescent p-modes in the photosphere are nonadiabatic.     

Properties of High-Frequency Wave Power Halos Around Active Regions: An Analysis of Multi-height Data from HMI and AIA Onboard SDO

We study properties of waves of frequencies above the photospheric acoustic cut-off of ≈5.3 mHz, around four active regions, through spatial maps of their power estimated using data from the Helioseismic and Magnetic Imager (HMI) and Atmospheric Imaging Assembly (AIA) onboard the Solar Dynamics Observatory (SDO). The wavelength channels 1600 Å and 1700 Å from AIA are now known to capture clear oscillation signals due to helioseismic p-modes as well as waves propagating up through to the chromosphere. Here we study in detail, in comparison with HMI Doppler data, properties of the power maps, especially the so-called “acoustic halos” seen around active regions, as a function of wave frequencies, inclination, and strength of magnetic field (derived from the vector-field observations by HMI), and observation height. We infer possible signatures of (magneto)acoustic wave refraction from the observation-height-dependent changes, and hence due to changing magnetic strength and geometry, in the dependences of power maps on the photospheric magnetic quantities. We discuss the implications for theories of p-mode absorption and mode conversions by the magnetic field.     

Mode Conversion of Solar <Emphasis Type="Italic">p</Emphasis>-Modes in Non-Vertical Magnetic Fields

Sunspots absorb and scatter incident f- and p-modes. Until recently, the responsible absorption mechanism was uncertain. The most promising explanation appears to be mode conversion to slow magnetoacoustic-gravity waves, which carry energy down the magnetic field lines into the interior. In vertical magnetic field, mode conversion can adequately explain the observed f-mode absorption, but is too inefficient to account for the absorption of p-modes. In the first paper of the present series we calculated the efficiency of fast-to-slow magnetoacoustic-gravity wave conversion in uniform non-vertical magnetic fields. We assumed two-dimensional propagation, where the Alfvén waves decouple. In comparison to vertical field, it was found that mode conversion is significantly enhanced in moderately inclined fields, especially at higher frequencies. Using those results, Cally, Crouch, and Braun showed that the resultant p-mode absorption produced by simple sunspot models with non-vertical magnetic fields is ample to explain the observations. In this paper, we further examine mode conversion in non-vertical magnetic fields. In particular, we consider three-dimensional propagation, where the fast and slow magnetoacoustic-gravity waves and the Alfvén waves are coupled. Broadly speaking, the p-mode damping rates are not substantially different to the two-dimensional case. However, we do find that the Alfvén waves can remove similar quantities of energy to the slow MAG waves.     

An experiment to observe the vertical gradient in line-of-sight velocity oscillations

Using intermediate degree p-mode frequency datasets for solar cycle 22, we find that the frequency shifts and magnetic indices show a `hysteresis' phenomenon. It is observed that the magnetic indices follow different paths for the ascending and descending phases of the solar cycle, as the descending path always seems to follow a higher track than the ascending one. However, for the radiative indices, the paths cross each other indicating phase reversal.     

Uncovering the Bias in Low-Degree p-Mode Linewidth Fitting

Obtaining reliable estimates of linewidths in the power spectra of low-degree p modes is problematic at low frequency. In this regime, the mode coherence time increases with decreasing frequency, often causing the modes to be unresolved in relatively long duration spectra. The signal-to-noise ratio is also less favourable at low frequency, resulting in fits to power spectra underestimating the true linewidth of the p modes owing to the tails of the Lorentzian peaks becoming dominated by the background noise. We use a numerical simulation approach to assess the effect of this bias on the fitted widths of p-mode peaks and calculate observational duration limits required to obtain an unbiased estimate of the p-mode linewidth as a function of frequency. This is done in four different cases, where the precision of the artificial data is set at 0.25, 0.50, 0.75, and 1.00 m?s^?1 by adding random?scatter to increase the sample standard deviation per 40-second measurement. In all cases, the observational duration required to accurately obtain width estimates increases beyond that required for sufficient spectral resolution below a certain threshold frequency. For modes at ≈?1500 μHz, with an amplitude of approximately ten times the background, observations of up to 972 days are required to obtain an unbiased estimate of the linewidth. This is equivalent to ≈?18 times the coherence time of the corresponding p modes.     

Enhanced p-Mode Absorption Seen Near the Sunspot Umbral – Penumbral Boundary

We investigate p-mode absorption in a sunspot using SOHO/MDI high-resolution Doppler images. The Doppler power computed from a 3.5-hour data set is used for studying the absorption in a sunspot. The result shows an enhancement in absorption near the umbral?–?penumbral boundary of the sunspot. We attempt to relate the observed absorption with the magnetic-field structure of the sunspot. The transverse component of the potential field is computed by using the observed SOHO/MDI line-of-sight magnetograms. A comparison of the power map and the computed potential field shows enhanced absorption near the umbral?–?penumbral boundary where the computed transverse field strength is higher.     

Solar Oscillation Frequency Changes on Time Scales of Nine Days

We establish that global solar p-mode frequencies can be measured with sufficient precision on time scales as short as nine days to detect activity-related shifts. Using ten years of GONG data, we report that mode-mass and error-weighted frequency shifts derived from nine days are significantly correlated with the strength of solar activity and are consistent with long-duration measurements from GONG and the SOHO/MDI instrument. The analysis of the year-wise distribution of the frequency shifts with change in activity indices shows that both the linear-regression slopes and the magnitude of the correlation varies from year to year and they are well correlated with each other. The study also indicates that the magnetic indices behave differently in the rising and falling phases of the activity cycle. For the short-duration nine-day observations, we report a higher sensitivity to activity.     

Mode Conversion of Solar p Modes in non-Vertical Magnetic Fields – i. two-Dimensional Model

Sunspots absorb incident p modes. The responsible mechanism is uncertain. One possibility is mode conversion to slow magnetoacoustic–gravity waves. In vertical field mode conversion can adequately explain the observed f-mode absorption, but is too inefficient to explain the absorption of p modes. In this investigation we calculate the efficiency of fast-to-slow magnetoacoustic–gravity wave conversion in non-vertical field. We assume two-dimensional propagation where the Alfvén waves decouple. It is found that resultant p-mode absorption is significantly enhanced for moderate inclinations at higher frequencies, whereas for p modes at lower frequencies, and the f mode in general, there is no useful enhancement. However, the enhancement is insufficient to explain the observed p-mode absorption by sunspots. Paper II considers the efficiency of mode conversion in non-vertical field with three-dimensional propagation, where fast and slow magnetoacoustic–gravity waves and Alfvén waves are coupled.     

The horizontal propagation of Pc1 pulsations in the ionosphere

The propagation properties of the various elements of the plane-wave angular spectrum of a Pc1 pulsation signal in the ionosphere are determined by a full-wave numerical analysis. A spectral component is characterized by the wave-vector azimuthal direction, and the Snell constant S. The isotropic R-mode transmission coefficient to ground is fairly flat for S ? 400, but thereafter (S > 500) drops rapidly with increasing S. Coupling of energy from the field-guided L-mode to the R-mode occurs along the entire length of the L-mode trajectory within the ionospheric duct in which the R-mode can propagate. Within the duct, the R-mode attenuation is determined largely by R to L-mode coupling, which is larger for E-W than for N-S azimuths, especially for steep angles of incidence (S < 100). This should lead to enhanced injection of energy into E-W high altitude, high velocity paths, but to higher E-W attenuation at oblique angles. For oblique propagation (S ? 200) horizontal group velocities are slightly higher than the Alfvén phase velocity at the F-layer peak, but about twice as high for steep angles (S ≈ 100).     

Multi-height Scattering Phase Shifts from Sunspots and Phase Differences Between Photospheric Heights with SDO/HMI and AIA Data

Following Couvidat (Solar Phys. 282, 15, 2013), we analyze data from the Helioseismic and Magnetic Imager (HMI) and the Atmospheric Imaging Assembly (AIA) instruments onboard the Solar Dynamics Observatory. Doppler velocity and continuum intensity at 6173 Å as well as intensity maps at 1600 Å and 1700 Å are studied on 14 active regions and four quiet-Sun regions at four heights in the solar photosphere. A Hankel–Fourier analysis is performed around these regions of interest. Outgoing–ingoing scattering phase shifts at a given atmospheric height are computed, as well as ingoing–ingoing and outgoing–outgoing phase differences between two atmospheric heights. The outgoing–ingoing phase shifts produced by sunspots show little dependence on measurement height, unlike the acoustic power reduction measured in Couvidat (2013). Phenomena happening above the continuum level, like acoustic glories, appear not to have a significant effect on the phases. However, there is a strong dependence on sunspot size, line-of-sight magnetic flux, and intensity contrast. As previously suggested by other groups, the region of wave scattering appears both horizontally smaller and vertically less extended than the region of acoustic power suppression, and occurs closer to the surface. Results presented here should help refine theoretical models of acoustic wave scattering by magnetic fields. Ingoing–ingoing phase differences between two measurement heights are also computed and show a complex pattern highly dependent on atmospheric height. In particular, a significant sensitivity of AIA signals to lower chromospheric layers is shown. Finally, ingoing–ingoing minus outgoing–outgoing phase differences between various measurement heights are discussed.     

Absorption of acoustic waves in monolithic and fibril sunspot models

It has been hypothesized that the observation of substantial absorption of acoustic power in the vicinity of sunspots may be explained by the transformation of acoustic oscillations into highly damped shear Alfvén waves in thin resonant layers. Analytical estimates of the efficiency of this process (Hollweg, 1988) are compared with direct one-dimensional numerical simulations of absorption by a magnetic barrier in a viscous medium. After slight modification, the estimate is found to give a good approximation to the numerical absorption rate.Further calculations are performed for scattering from a magnetic field of fibril structure. Such models are better able to explain the spatial structure of the absorbing region implied by the observations. It is found that the existence of a multiplicity of surfaces at which resonant absorption occurs can considerably increase the total energy absorption coefficient. Resonant effects involving the multiple reflection of acoustic waves within such structures can also lead to enhanced absorption. Fibril models, therefore, produce significantly increased absorption over a wide range of plausible parameter values, and are a more plausible explanation for the observed p-mode scattered power deficit than resonant absorption in a monolithic structure.     

Empirical estimate of p-mode frequency shift for solar cycle 23

We have obtained empirical relations between the p-mode frequency shift and the change in solar activity indices. The empirical relations are determined on the basis of frequencies obtained from BBSO and GONG stations during solar cycle 22. These relations are applied to estimate the change in mean frequency for the cycle 21 and 23. A remarkable agreement between the calculated and observed frequency shifts for the ascending phase of cycle 23, indicates that the derived relations are independent of epoch and do not change significantly from cycle to cycle. We propose that these relations could be used to estimate the shift in p-mode frequencies for past, present and future solar activity cycles, if the solar activity index is known. The maximum frequency shift for cycle 23 is estimated to be 265±90 nHz, corresponding to a predicted maximum smoothed sunspot number 118.1±35.     

How Good are the Predictions for Oscillation Frequencies

We have used available intermediate degree p-mode frequencies for solar cycle 23 to check the validity of previously derived empirical relations for frequency shifts (Jain et al., 2000). We find that the calculated and observed frequency shifts during the rising phase of cycle 23 are in good agreement. The observed frequency shift from minimum to maximum of this cycle as calculated from MDI frequency data sets is 251±7 nHz and from GONG data is 238±11 nHz. These values are in close agreement with the empirically predicted value of 271±22 nHz.     

Toward Waveform Heliotomography: Observing Interactions of Helioseismic Waves with a Sunspot

We investigate how helioseismic waves that originate from effective point sources interact with a sunspot. These waves are reconstructed from observed stochastic wavefields on the Sun by cross-correlating photospheric Doppler-velocity signals. We select the wave sources at different locations relative to the sunspot, and investigate the p- and f-mode waves separately. The results reveal a complicated picture of waveform perturbations caused by the wave interaction with the sunspot. In particular, it is found that for waves originating from outside of the sunspot, p-mode waves travel across the sunspot with a small amplitude reduction and slightly higher speed, and wave amplitude and phase get mostly restored to the quiet-Sun values after passing the sunspot. The f-mode wave experiences some amplitude reduction passing through the sunspot, and the reduced amplitude is not recovered after that. The wave-propagation speed does not change before encountering the sunspot and inside the sunspot, but the wavefront becomes faster than the reference wave after passing through the sunspot. For waves originating from inside the sunspot umbra, both f- and p-mode waves show significant amplitude reductions and faster speed for all propagation paths. A comparison of positive and negative time lags of cross-correlation functions shows an apparent asymmetry in the waveform changes for both the f- and p-mode waves. We suggest that the waveform variations of the helioseismic waves interacting with a sunspot found in this article can be used for developing a method of waveform heliotomography, similar to the waveform tomography of the Earth.     

Spatio-Temporal Analysis of Photospheric Turbulent Velocity Fields Using the Proper Orthogonal Decomposition

The spatio-temporal dynamics of the solar photosphere are studied by performing a proper orthogonal decomposition (POD) of line-of-sight velocity fields computed from high-resolution data coming from the SOHO/MDI instrument. Using this technique, we are able to identify and characterize the different dynamical regimes acting in the system. All of the POD modes are characterized by two well-separated peaks in the frequency spectra. In particular, low-frequency oscillations, with frequencies in the range 20?–?130 μHz, dominate the most energetic POD modes (excluding solar rotation) and are characterized by spatial patterns with typical scales of about 3 Mm. Patterns with larger typical scales, of about 10 Mm, are dominated by p-mode oscillations at frequencies of about 3000 μHz. The p-mode properties found by POD are in agreement with those obtained with the classical Fourier analysis. The spatial properties of high-energy POD modes suggest the presence of a strong coupling between low-frequency modes and turbulent convection.     

Modelling p-Mode Interaction with a Spreading Sunspot Field

Sunspots absorb and scatter incident p modes. The dominant mechanism is still uncertain. One possibility, mode conversion to slow magneto-acoustic waves, has been shown to yield results in agreement with observations for the f mode only. Absorption of p modes in simple vertical magnetic field models is too weak by an order of magnitude or more. Here we report on numerical calculations of p modes encountering a simple sunspot model with field which spreads with height. It is found that p-mode absorption is greatly enhanced by field spread, to a level consistent with observations, and it appears that it occurs preferentially in the outer regions of the spot, in line with recent results from acoustic holography.     

Observation of hysteresis between solar activity indicators andp-mode frequency shifts for solar cycle 22

Using intermediate degreep-mode frequency data sets for solar cycle 22, we find that the frequency shifts and magnetic activity indicators show a &amp;#x201C;hysteresis&amp;#x201D; phenomenon. It is observed that the magnetic indices follow different paths for the ascending and descending phases of the solar cycle while for radiative indices, the separation between the paths are well within the error limits.