Doppler Events in the Solar Photosphere: The Coincident Superposition of Fast Granular Flows and p-Mode Coherence Patches

Observations of the solar photosphere show spatially compact large-amplitude Doppler velocity events with short lifetimes. In data from the Imaging Magnetograph eXperiment (IMaX) on the first flight of the Sunrise balloon in 2009, events with velocities in excess of 4σ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\sigma $\end{document} from the mean can be identified in both intergranular downflow lanes and granular upflows. We show that the statistics of such events are consistent with the random superposition of strong convective flows and p-mode coherence patches. Such coincident superposition complicates the identification of acoustic wave sources in the solar photosphere, and may be important in the interpretation of spectral line profiles formed in solar photosphere.


Introduction
Spectropolarimetric inversion of data from the Imaging Magnetograph eXperiment (IMaX) instrument (Martínez Pillet et al., 2011) on the first flight of the SUNRISE stratospheric balloon (Solanki et al., 2010)  Two physically distinct processes contribute to the magnitude of the Doppler velocity observed in the solar photosphere. Thermal convection is visible as granulation, with characteristic Doppler velocity amplitudes between ±0.5 -1.5 km s −1 (Title et al., 1989), and the random superposition of the solar acoustic oscillations (Ulrich, 1970) with maximum individual mode amplitudes of about ±15 cm s −1 (e.g., Christensen-Dalsgaard, 2003) producing coherence patches with median vertical velocity amplitude of about ±0.5 km s −1 and maximum amplitudes on the order of ±1 km s −1 (Leighton, Noyes, and Simon, 1962;Libbrecht, 1988). In this paper we study these two components of the flow at high spatial resolution and show how together they yield the observed Doppler flashes.
In Section 2 we describe the SUNRISE I IMaX data in more detail and discuss the criteria used to identify the extreme Doppler velocities events. In Section 3 we separate granular and p-mode contributions to the Doppler maps, examine the two components individually, and show that the number and amplitude distribution of extreme events are statistically consistent with the random superposition of locally strong granular flows and p-mode coherence patches. Implications are discussed in Section 4.

Observations
The SUNRISE I stratospheric balloon mission observed the Sun for 130 hours in 2009. The balloon carried a 1 m Gregorian telescope with an effective focal length of ∼ 25 m at a float altitude of 35 -40 km (Solanki et al., 2010). The post-focus IMaX instrument used two custom nematic liquid crystal variable retarders, a commercial polarizing beamsplitter, and a double-pass LiNbO3 etalon (Álvarez-Herrero et al., 2006) to measure all four Stokes parameters within the Zeeman-sensitive Fe I 525.02 nm line and in the neighboring continuum over a 50 × 50 arcsecond field of view with a cadence between 10 and 33 s (Martínez Pillet et al., 2011). We present results from measurements taken at four spectral line positions (−80, −40, +40, and +80 mÅ) and in the neighboring continuum (+227 mÅ). Two regions near solar disk center were observed with spectral resolution of 85 mÅ at a cadence of 33 seconds, one for a total of 23 minutes and the other for 32 minutes (hereafter called image Set 1 and image Set 2, respectively). A Milne-Eddington inversion (MILOS-Orozco Suárez et al., 2007; was used to determine the three magnetic field components, lineof-sight velocity, and plasma temperature over the field of view of each image frame in the two time series. Figure 1a displays the line-of-sight velocity in a single frame. Animations of the full time series are available as supplementary material. The region under consideration is free of any large-scale magnetism (see Figure 1b). Granulation is readily discernible, and acoustic oscillations are apparent as a time-dependent wavering across the field of view in the animated time series and as large regions (∼ 2 × 2 Mm 2 patches) of enhanced or depressed vertical velocities in the still image of Figure 1a.

Event Definition
Spatially compact large-amplitude Doppler velocity events with a duration much shorter than the lifetime of granules are observed at apparently random intervals and locations in the image time series. These events occur with both signs; downflow events appear as striking flashes in the intergranular lanes and upflow events appear as high-velocity localized sites within granules. With the color table selected for Figure 1a, the events can be seen as compact bright white regions in the intergranular lanes (e.g., at (x, y) = (2, 24) in Figure 1a) and bright yellow areas in the granules (e.g., at (x, y) = (25, 15) in Figure 1a). For analysis, we defined a Doppler event to be any region with an area of at least 10 pixels 2 (∼ 400 km 2 ) that shows velocities in excess of 4σ above or below the temporal and spatial mean over three or more consecutive frames. This corresponds to velocities below −2.65 or above 1.98 km s −1 in time-series Set 1 and below −2.71 or above 2.11 km s −1 in Set 2. The average event rate across the two time series is 1.9 ± 0.3 upflow events and 2.0±0.3 downflow events per frame over the 30 × 30 Mm 2 central subregion retained after the apodization we employed in the Fourier filter described below.

Convective and Acoustic Contributions
As discussed earlier, there are two contributions to the Doppler velocity measured at any position in the solar photosphere: the convective flow, and the acoustic oscillations. We separated these two contributions by applying a Fourier filter to the Doppler data (e.g., Hill, 1988;Schou et al., 1998). The images were spatially apodized using a two-dimensional cosine-bell taper of width 39 pixels applied to each edge, Fourier transformed in time and space, and separated into the granular and p-mode components either side of 5 km s −1 (see Figure 2). Single frames of the resulting granular and acoustic contributions are shown in Figure 1c and 1d.
The distributions of Doppler velocities measured for each component are shown in Figure 3. Not surprisingly, the p-mode contribution is normally distributed about zero, since it results from the random superposition of oscillatory symmetric eigenfunctions. On the other hand, the distribution of the convective velocities is significantly non-Gaussian, with enhanced occurrence of large amplitude flows of both signs. The mean is shifted to negative values because the area of the upflowing regions exceeds that of the downflows, and the distribution is asymmetric, with extreme values more prevalent in downflowing regions (positive values in Figure 3). The median upflow speed (0.60 km s −1 for Set 1 and 0.61 km s −1 for Set 2) is higher than the median downflow speed (0.43 km s −1 for Set 1 and 0.45 km s −1 for Set 2), but the upflow speed distribution is much narrower than the downflow speed distribution. While the peak magnitude of downflow speeds (∼ 8 km s −1 ) is higher than the peak magnitude of upflow speeds (∼ 6 km s −1 ), see Figure 3, we note the granular upflows are highly structured with peak upflow speeds in small, compact regions approaching those found in downflow lanes.

Coincident Superposition
To investigate the origin of the extreme Doppler velocity events identified, we tested the hypothesis that such events are caused by the random superposition of the p-mode and granular contributions. We created synthetic datasets by superimposing the observed p-mode and granulation fields, spatially offsetting the two components and coadding them to form synthetic-image time series. Each frame capturing the p-mode contribution was periodically shifted in increments of 20 pixels 32 times, separately horizontally and vertically and simultaneously in both directions. This process created 102 synthetic observations at each time  step. Since each frame is over 700 pixels on a side and the minimum event size is taken to be 10 pixels 2 , shifts of 20 pixels avoid double-counting synthetic events.

Event Amplitude Distributions
The model and the original data Doppler amplitude distributions are quite similar (Figure 4), with some small differences apparent in the distribution shoulders and more noise in the observed distribution tails than in the synthetic data for which more time series are available. More constraining than a comparison between the velocity distributions is an assessment of Figure 5 Probability density of the number of event counts per frame in the observations and the random superposition model. event rates, however, since this captures spatial and temporal coherences within the Doppler map via the event definition.
The overall mean event rates are listed in Table 1 of the Appendix. The observed mean event rate lies within 2σ of the mean rate in the synthetic time series produced by the random superposition of granular and p-mode motions. The probability densities of the number of counts per frame ( Figure 5) also agrees quite well with that of the observed data, but because of the low number of total counts in the observed data, it is difficult to tell whether any differences are significant.

Occurrence Frequency
To examine this further, we considered the probability of the number of events observed in each frame. Since event counts are small, the distribution approximates a Poisson distribution, so we calculated for each frame the Poisson probability of finding the number of events observed if the mean is that determined from the random superposition model. Figure 6a plots that probability as a function of frame number for the observed time series, along with the distributions of those values for upflow and downflow events in both simulations projected along the y-axis. Mean Poisson probability values are indicated with solid horizontal fiducial lines. Similarly, in Figure 6b we plot the distribution of the Poisson probability of finding the specific number of events in each frame of a synthetic dataset given the mean number of event counts from all the other model sets. In this way, each model time series is treated in the same manner as the observational time series.
The mean values of the Poisson probabilities of the observed counts (indicated in Figure 6b by solid vertical fiducial lines) falls at or within 1σ of the means of the synthetic data Poisson probability distributions. This is true for both upflows and downflows in all observational time series. We conclude that the number of outstanding Doppler events in each frame of the observational data is statistically consistent with counts that would be achieved by the random superposition of p-mode and the granular flows.

Conclusion
The random superposition of solar acoustic oscillation coherence and underlying convective flows can cause intermittent Doppler signals that have amplitudes in excess of 4σ from the mean, values that are thus faster than 99.7% of those observed if the distribution is approximated as Gaussian. These Doppler events occur in both upflowing and downflowing regions, and flows of this magnitude are three times more likely to occur in an unfiltered time series than in one from which the p-modes have been removed. It is important to note that these events are real physical occurrences, not an artifact of the observations. Their existence may be critical to the interpretation of spectral lines, the identification of solar acoustic sources, and the understanding of jet-like structures. They may also cause confusion in interpreting observations as instances of specific dynamical mechanisms, particularly when observational time series are too short to adequately remove the p-mode contributions. Set 2 1.7 ± .1/frame 5.70 × 10 −5 ± 1% Downflows Set 1 2.0 ± .2/frame 6.87 × 10 −5 ± 1% Set 2 1.8 ± .1/frame 6.00 × 10 −5 ± 1%