Microjets in the penumbra of a sunspot

TL;DR

This study presents an automated analysis of Penumbral Microjets (PMJs) in sunspots, revealing their statistical properties, spectral profiles, and repeated hotspot locations, supporting their heating to transition region temperatures.

Contribution

It introduces an automated detection method for large PMJ samples and provides detailed spectral and statistical analysis, improving understanding of PMJ characteristics.

Findings

453 tracked PMJ events identified

Average length 640 km, lifetime 90 s

Spectral profiles show enhanced inner wings and brighter line core

Abstract

Penumbral Microjets (PMJs) are short-lived jets found in the penumbra of sunspots, first observed in wide-band Ca H-line observations as localized brightenings, and are thought to be caused by magnetic reconnection. Earlier work on PMJs has been focused on smaller samples of by-eye selected events and case studies. It is our goal to present an automated study of a large sample of PMJs to place the basic statistics of PMJs on a sure footing and to study the PMJ Ca II 8542 Angstrom spectral profile in detail. High spatial resolution and spectrally well-sampled observations in the Ca II 8542 Angstrom line obtained from the Swedish 1-m Solar Telescope (SST) are reduced by a Principle Component Analysis and subsequently used in the automated detection of PMJs using the simple learning algorithm k-Nearest Neighbour. PMJ detections were verified with co-temporal Ca H-line observations. A total of 453 tracked PMJ events were found, or 4253 PMJs detections tallied over all timeframes and a detection rate of 21 events per timestep. From these, an average length, width and lifetime of 640 km, 210 km and 90 s were obtained. The average PMJ Ca II 8542 Angstrom line profile is characterized by enhanced inner wings, often in the form of one or two distinct peaks, and a brighter line core as compared to the quiet Sun average. Average blue and red peak positions were determined at -10.4 km/s and +10.2 km/s offsets from the Ca II 8542 Angstrom line core. We found several clusters of PMJ hotspots within the sunspot penumbra, where PMJ events occur in the same general area repeatedly over time. Our results indicate smaller average PMJs sizes and longer lifetimes compared to previously published values, but with statistics still in the same orders of magnitude. The investigation and analysis of the PMJ line profiles strengthen the proposed heating of PMJs to transition region temperatures.

Figures

Tables

Table 1. Table 1: PMJ detection statistics

Total PMJ detections (over all frames) [count]	4253
Tracked PMJ objects [count]	453
Mean PMJ detections per frame $\left[\frac{\text{\# events}}{\text{frame}}\right]$	21
Mean lifetime,tracked PMJs (all) [s]	117
Mean lifetime, tracked PMJs (< 8 min)${}^{\text{a}}$ [s]	90
Mean length${}^{\text{b}}$ [km]	640
Mean width${}^{\text{b}}$ [km]	210
PMJ minimum position${}^{\text{c}}$ [km s-1]	0.14
PMJ blue peak offset${}^{\text{c}}$ [km s-1]	$-10.4$
PMJ red peak offset${}^{\text{c}}$ [km s-1]	$10.2$

Table 2. Table 2: Summary of PMJ peak-presence in individual profiles

Nr. of (9-pixel average) profiles
used in peak detection	3953
Nr. of peaks detected	1868
Peak combinations	Nr. of profiles
in profiles
Blue and/or red peak	1780
Both blue and red peak	114
Blue peak (with possible red)	1725
Only blue peak	1637
Red peak (with possible blue)	143
Only red peak	29

Full text

*Aims. *Earlier work on PMJs has been focused on smaller samples of by-eye selected events and case studies. It is our goal to present an automated study of a large sample of PMJs to place the basic statistics of PMJs on a sure footing and to study the PMJ Ca ii 8542 Å spectral profile in detail.

*Methods. *High spatial resolution and spectrally well-sampled observations in the Ca ii 8542 Å line obtained from the Swedish 1-m Solar Telescope (SST) are reduced by a Principle Component Analysis and subsequently used in the automated detection of PMJs using the simple learning algorithm k-Nearest Neighbour. PMJ detections were verified with co-temporal Ca ii H line observations.

Results. A total of 453 tracked PMJ events were found, or 4253 PMJs detections tallied over all timeframes and a detection rate of 21 events per timestep. From these, an average length, width and lifetime of 640 km, 210 km and 90 s were obtained. The average PMJ Ca ii 8542 Å line profile is characterized by enhanced inner wings, often in the form of one or two distinct peaks, and a brighter line core as compared to the quiet Sun average. Average blue and red peak positions were determined at $-10.4$ km s-1 and $+10.2$ km s*-1* offsets from the Ca ii 8542 Å line core. We found several clusters of PMJ hotspots within the sunspot penumbra, where PMJ events occur in the same general area repeatedly over time.

*Conclusions. *Our results indicate smaller average PMJs sizes and longer lifetimes compared to previously published values, but with statistics still in the same orders of magnitude. The investigation and analysis of the PMJ line profiles strengthen the proposed heating of PMJs to transition region temperatures. The presented statistics on PMJs form a solid basis for future investigations and numerical modeling of PMJs.

1 Introduction

Penumbral microjets (PMJs) are short-lived, elongated, transients in the chromosphere of sunspot penumbrae. They were discovered (Katsukawa et al. 2007) in Ca II H time sequences from Hinode’s 3 Å wide imaging filter in which PMJs display a 10–20% brightness enhancement as compared to surrounding penumbral structures. In the Hinode observations, PMJs have typical lifetimes of up to 1 minute, lengths between 1000 and 4000 km, widths of about 400 km, and apparent rise velocity faster than 100 km -1 (Katsukawa et al. 2007).

Penumbrae are known to host strong convectively driven plasma flows and magnetic fields that vary significantly at small spatial scales, both in inclination and magnitude (see e.g., Borrero & Ichimoto 2011). In this magnetically stressed environment, magnetic reconnection appears a viable candidate as driver of PMJs. This is supported by the measurements of the apparent inclination of PMJs with respect to the photospheric penumbral filaments (Katsukawa et al. 2007), and magnetic fields (Jurčák & Katsukawa 2008). Further indications of the reconnection scenario come from Katsukawa & Jurčák 2010 who reported the association of small photospheric downflow patches with some PMJs. These could be interpreted as the downward flows from magnetic reconnection above the photosphere. Evidence of progressive heating along PMJs was reported by Vissers et al. 2015 (Fig.5), who found clear responses in Mg ii k, C ii, and Si IV slit-jaw images of IRIS to PMJs observed in Ca II lines. C ii and Si IV emission towards the top of PMJs suggests heating to transition region temperatures.

Reardon et al. 2013 study transients in a sunspot penumbra from spectral imaging data in the Ca II 8542 line. Aided by co-temporal Hinode Ca II H imaging, they identified several PMJs in their dataset. The Ca II 8542 line profiles show enhanced emission in the wings out to $\pm 0.5$ Å, with peaks at about $\pm 0.3$ Å, and a line core that shows little difference as compared to the surroundings. They point at the similarity with Ca II 8542 spectral profiles of Ellerman bombs, like for example as shown in Vissers et al. 2013.

In this study, we expand on the observational characterization of PMJs analyzing a high-spatial resolution time series of both narrow-band (1 Å) Ca II H filtergrams and spectral imagery in Ca II 8542. We employ an automated detection scheme to built a large statistical sample of PMJs. The detection scheme neatly takes advantage of the many sampling position in the Ca II 8542 line, utilizing the full line profile. The dimensionality of the observations in wavelengths positions is first reduced employing Principal Component Analysis (PCA), and is then used in detections performed employing the k-Nearest Neighbour algorithm, and finally followed by object tracking and statistical analysis.

2 Observations

Active region AR11084 was observed on 28-June-2010 with the Swedish 1-m Solar Telescope (SST, Scharmer et al. 2003a) on La Palma. The field of view was centered on the near-circular sunspot with fully developed penumbra at heliocentric coordinates $(X,Y)=(710,-339)$ ($\mu=\cos\theta=0.55$, with $\theta$ the observing angle). The seeing was excellent for the full 41 min duration of the time series (starting at 09:18:29 UT), and the image quality further benefited from the adaptive optics system (Scharmer et al. 2003b) and image reconstruction with the Multi-Object Multi-Frame Blind Deconvolution method (MOMFBD, van Noort et al. 2005). We analyzed data from instruments on both branches of the optical beam: from the CRisp Imaging SpectroPolarimeter (CRISP, Scharmer et al. 2008) on the long-wavelength branch “red beam”), and filtergram imaging in Ca ii H on the short-wavelength branch (“blue beam”).

With CRISP, we sampled the Ca ii 8542 Å line at 37 line positions, with equidistant 55 mÅ steps out to $\pm 880$ mÅ, and additional sampling at $\pm 948$ and $\pm 1034$ mÅ. CRISP has a FWHM of 110 mÅ at 8542 Å, so the Ca ii line is critically sampled throughout the central part of the spectral line profile. We acquired 8 exposures per spectral sampling which were used for MOMFBD image reconstruction. In addition, single wavelength spectro-polarimetric samplings of the Fe i 6302 Å line were acquired. Unfortunately, due to erroneous calibration settings, the precise wavelength for this sampling was unknown which resulted in noisy and effectively useless maps of the four Stokes parameters. The acquisition time for Ca ii 8542 Å was 8.1 s and the temporal cadence of the time series was 12.4 s. After MOMFBD restoration of the individual spectral line scans, the data was put together as a time series after correction for the CRISP prefilter (FWHM 9.3 Å for Ca ii), compensation of the diurnal field rotation, rigid alignment, and destretching.We used early versions of the different procedures that were later put together as the reduction pipeline for CRISP data (de la Cruz Rodríguez et al. 2015) including the post-MOMFBD correction for remaining small-scale seeing deformations due to the non-simultaneity of the sequentially recorded narrowband CRISP images (Henriques 2012). The effective field of view of the time series is 55″$\times$ 55″, with a pixelscale of 0$\aas@@fstack{\prime\prime}$059 per pixel.

In the blue beam, synchronized filtergrams were recorded at a rate of 10.8 frames s*-1* in the Ca ii H line core (filter FWHM 1.1 Å) and with a wider passband filter (FWHM 10 Å) at $\lambda=3954$ Å, between the Ca ii H and K lines. These two imaging channels were MOMFBD restored to produce a time series with an effective cadence of half the CRISP data, 6.2 s. The alignment to the CRISP data was done by cross-correlation of the red and blue wideband channels, which both show the photosphere.

Figure 1 shows example frames from the observations, displaying the sunspot at offsets $-1032$ mÅ  and $-275$ mÅ in the Ca ii 8542 Å  line. Also displayed are the nominal borders of the umbra and penumbra as well as the area over which the average quiet sun Ca ii 8542 Å  line-profile used for comparison was computed over. A cropped image of the observations also showcases a strong inverse Evershed flow.

3 Methods

Here a brief overview of the used methods is given. Prior to the employment of the pipeline described below, the observations were investigated using the CRisp SPectral EXplorer (CRISPEX Vissers & Rouppe van der Voort 2012), which was used to interactively browse the MOMFBD reduced observations. CRISPEX was also instrumental in the by-eye assembly of the reference set of PMJs and non-PMJ objects in the observations for later employment of the k-NN algorithm.

We restrict to a qualitative description and focus only on the basic concepts and overall structure. For an in-depth treatment, the reader is directed to Drews (2014), Section 5, in which the full methodology for the automated detection scheme and detection process is described, as well as a full explanation of the subsequent object-tracking.

The main working steps, starting from the post-MOMFBD SST observations, can be summarized in the following discrete steps:

Preliminary identification of PMJs in Ca ii 8542 Å  using Ca ii H line observations as reference 2. 2.

Principle Component Analysis: dimensionality reduction and data compression 3. 3.

Detection of PMJs using the k-Nearest Neighbour algorithm 4. 4.

Object tracking and statistical analysis

These four distinct steps are covered in some more detail in the subsections below.

3.1 Preliminary identifications of PMJs

To justify the claimed observation of PMJs in the Ca ii 8542 Å  line, a subset of observations was first compared to by-eye detections of PMJs in the co-observed Ca ii H line, as the detection of PMJs and their appearance is firmly established in this line (Katsukawa et al. 2007, Jurčák & Katsukawa 2008, Katsukawa & Jurčák 2010 and Jurčák & Katsukawa 2010).

Earlier observations of PMJs in Ca ii 8542 Å were presented by Reardon et al. 2013, Drews (2014), as well as by Vissers et al. 2015.

A quick qualitative study was enough to show that many, if not most, Ca ii H PMJ detections have spatially coinciding similar features in the Ca ii 8542 Å  line, in particular slightly blue-ward of the nominal line center wavelength. Similar as in Ca ii H, PMJs appear in selected spectral positions in the Ca ii 8542 Å line as short-lived, elongated brightenings in the sunspot penumbra.

This is illustrated in Fig. 2, which shows four example images in the observations at the same timeframe. One image is in the Ca ii H line, one in the Ca ii 8542 line core and two in the Ca ii 8542 Å line at an offset of $-275$ mÅ. PMJs proved to be most visible in by-eye detections in the Ca ii 8542 Å line in images at an offset of $-275$ mÅ. The arrows in the four panels all point to the same pixel positions, and it is clear that PMJ features are present in both diagnostics (the Ca ii H line and the Ca ii 8542 Å line scan). This is especially true when comparing features present in the Ca ii H line  which are clearly visible in the $-275$ mÅ Ca ii 8542 Å  line offset images as well. Detection borders of PMJs overplotted in panel (d) highlight that most by-eye selected examples in this frame were caught by the automated detection scheme that will be presented below (see Sect. 3.2 - 3.4.3). Notably, two PMJs as selected by-eye on the right of the FOV are classified as one event by the automated scheme. Further, one of the events is not detected by the detection scheme, marked by the second arrow from the left (manual inspection using CRISPEX indicate that this event’s spectral profile is not very distinct over a larger area).

From these examples, and other qualitative inspections of PMJs that visually coincided spatially in the Ca ii H line as well as in the Ca ii 8542 Å  line observations, the assumption that these events are the same physical objects is validated. Further investigation of PMJs in the Ca ii 8542 Å  line and the ascertainment of a distinct spectral line profile is therefore warranted.

3.2 Principle Component Analysis

The detections of PMJs in the presented automated approach is based on their distinct line profiles in the Ca ii 8542 Å  line. Intensity differences throughout the field of view, due to both limb-darkening as well as local variations, proved to make detections using the k-NN algorithm difficult. Intensity variations affecting the whole line profile may not impact the shape of the given pixel’s line profile, but may shift the overall intensity, including key sampling points where PMJs are visible, such that the similarity measures in the algorithm were less meaningful. For this reason, prior to PCA treatment, the observations were first normalized. The normalization consisted of normalizing each line profile in all pixels in all timeframes to its own sum. This preserved the line profile shapes, but effectively removed any overall intensity variations. The preservation of line profiles shapes consequently carried over into the PCA treated observations used in the k-NN detection scheme.

PCA is both a method for data analysis, as well as being a useful tool for data compression. In Shlens (2014) an introductory overview to PCA is given.

In PCA, the covariance between variables in a dataset is computed, constructing its covariance matrix. Subsequently, the associated eigen-vectors and -values of this matrix are found. The eigenvectors will correspond to a new set of basis vectors along which the data can be projected, yielding a linearly independent dataset with no cross-correlation between variables. Normalizing the associated eigenvalues, these yield the relative contribution of the new basis vector variables to the total variance of the newly aligned dataset. In practice, this variance can be equated to the informational contribution from the given new eigenvector. This presents the opportunity for compression of the dataset, as discarding variables along eigenvectors with low informational value will not yield significant information loss overall, and additionally, this loss will be quantifiable.

In the present context, the 37 wavelength sample points of the Ca ii 8542 Å  line are treated as the variables in a 37-dimensional dataset. Hence, this is the dataset that is analyzed and compressed using PCA as outlined above.

For the present pipeline, the computation of the covariance matrix of the Ca ii 8542 Å  line observations was performed following Bennet et al. (2009).

Here, a numerically stable single-pass algorithm is presented. It provides the benefit of a lighter numerical workload as a naive (but generally stable) approach requires two passes over the dataset. In such an approach the mean of a dataset is first computed, followed by the computation of the needed powers of this mean in the second pass.

The employed single-pass of Bennet et al. (2009) also avoids common numerical pitfalls in the computation of the covariances in terms of numerical instabilities when calculated in a single-pass approach.

Using the found eigenvectors and corresponding eigenvalues, it was determined that the variables along 7 eigenvectors describe the original observations to an accuracy, or informational content, of 97%, which was deemed acceptable, whilst yielding a compression of the data to 19% of the original size. Thus, these 7 eigenvectors were chosen as the Principle Components of the dataset, and the observations aligned and compressed along them.

Different morphological features, all with distinct spectral profiles (e.g., long fibrils, umbral flashes, fibrils with strong flows) are clearly identifiable in maps of the different Principle Components. This lends intuitive credence to the PCA reduction method, in that the different “new spectral” variables still represent real features picked out from the original sampling positions. For examples, we refer to Drews (2014), Section 5.

3.3 The k-Nearest Neighbour algorithm

The k-Nearest Neighbour algorithm is conceptually simple, yet powerful. This is one of the reasons it is widely applied in signal processing tasks such as facial and voice recognition as well as machine reading. For an introduction to the specifics of the algorithm, an overview and discussion for improvement of the algorithm is given in Guo et al. (2004). Further, Yang & Liu (1999) compare the algorithm to other classifiers.

Often termed the simplest of the learning algorithms, the k-NN algorithm is based on a comparative approach in which data to be classified is related to a pre-classified reference set of the same type. In practice, this means that a reference set is assembled using expert knowledge or a manual classification scheme, which is then used to classify the rest of the data using a similarity measure. Here, a simple euclidean metric is employed in the 7-dimensional PCA reduced dataset. A reference set was assembled using by-eye detections, noting the temporal and spatial location in the observations. At present, a reference set corresponding to a total number of 958 positions in time and space was assembled. This reference set is further divided into 168 PMJ positions (55 separate events) and 790 background positions of large diversity. These specific numbers were found to yield robust results following a trial-and-error approach, studying the number and variability of reference events needed until the results were satisfactory and consistent. As the reference set is polled for each automatic identification, an unnecessarily large reference set is to be avoided.

The background positions correspond to datapoints that are clearly not PMJs. The k-NN algorithm uses both object and non-object entries in the reference set for identification - for identification each vector in the PCA reduced data has its associated distance computed to all points in the reference set. A number of k-Nearest Neighbours are then polled, and the point is classified by majority vote. The selection of the parameter k is performed using an accuracy test of cross-classification of the assembled reference set. The classifications resulted in binary maps for each timeframe, consisting of background and PMJ detections which were then further processed, as outlined in Sect. 3.4. Figure 3 shows the Ca ii 8542 Å line profiles of the individual PMJs and background, or non-PMJ positions, that were assembled for the kNN reference set. It bears remarking that these profiles are separated for the sake of clarity for the two plots, but form one reference set in practice, with profiles marked as PMJ or non-PMJ in the set when polled. The selected PMJ profiles have quite the span in terms of intensity throughout the line, but are well-defined by their shape. This is also illustrated by their included average, which presents itself as a very well defined PMJ-like profile (which will be made clear in Sect. 4). The non-PMJ profiles span a wide variety of profiles, sampling positions in the quiet Sun, penumbra, umbra as well as features such as the strong inverse Evershed flow present in the observations.

3.4 Object tracking, statistical analysis and extraction of the Ca ii 8542 Å  line-profile

The post-processing of the binary maps of raw PMJ detections output by the k-NN algorithm was performed in several steps, which are described below. First, the binary maps were cleaned of noise and objects tracked through time, see Sect. 3.4.1. Following this, base statistics were extracted from the basis of these tracked objects, see Sect. 3.4.2. Lastly, line-profiles in the Ca ii 8542 Å  line were found, and this process is detailed in Sect. 3.4.3.

3.4.1 Noise removal and object tracking

To remove spurious signals, the binary detections output from the k-NN algorithm were first run through an 8-pixel connectivity mask, removing lone falsely identified pixels.

Further, a lower “hard” area limit of 20 pixels was implemented on object sizes. Additionally, a “soft” area limit of 50 pixels was also implemented. This means that any areas in the original bitmaps below a size of 20 pixels total were not considered at all. The soft limit meant that when during tracking no area of a given tracked object reached a size of 50 pixels, the tracked areas were discarded. This means that tracked objects could have sizes of below 50 pixels, but only as long as the given object reached a size as large, or larger, than 50 pixels in at least one timeframe. This was meant to ensure that it was possible to catch the smaller beginnings of objects, but not track spurious signals that never developed and to avoid noise from being tracked for long periods.

These area limits would correspond to cubic areas with sides $4.47$ pixels $=194$ km and $7.07$ pixels $=307$ km for the hard and soft limit respectively, though since a fitting of ellipse-shapes is later performed (see below) the spatial limits along one axis may be smaller. The limits given may seem prohibitively large, but are chosen after trial and error and yield reasonable results. The final limits were chosen since limits with lower values allowed small-scale noise in the bitmaps to be tracked and erroneously labeled as PMJs. This noise was mostly caused by a prominent inverse Evershed flow (see Fig. 1) in the lower right of the observations.

The object tracking was performed using relatively simple measures of spatial distances in the observations to relate discrete detections across timeframes. Here, a maximum distance between center-of-mass pixels of $15$ pixels $=652$ km between timeframes was used. This is equivalent to a projected speed of approximately $52$ km -1, though it must be kept in mind that due to the morphing of detection areas (and thus shifts of center-of-mass pixel positions) between frames this threshold can not be seen as a direct threshold on real movement of detected PMJs. Detections were also restricted to an outline of the photospheric penumbra, as seen in the wings of the Ca ii 8542 Å  line, see Fig. 1. Only objects with center-of-mass pixels inside this outline were included in the subsequent analysis. Further, spurious detections inside an outline of the umbra were also discarded, also see Fig. 1 (these events were most likely related to umbral flashes, as was evident when investigating using CRISPEX).

3.4.2 Basic statistics extraction

A statistical analysis on the resultant final PMJ detections was then carried out. For estimations of detected object sizes, a fitting of ellipses on the detected objects was carried out. The reasoning behind this was the assumption that jet-like objects like PMJs should in principle exhibit a somewhat elongated shape. The tracking was thus restricted to PMJs corresponding to best-fit-ellipses with an eccentricity of $0.9$, corresponding of an approximate ratio of semi-major to semi-minor axis in the ellipses of 2.3:1. The associated major and minor axes of these fitted ellipses for each single object were employed as estimates for the lengths and widths of the given detections respectively (see Fig. 4 for examples of these fitted ellipses). This allowed for the collection of size-statistics. Angular positions around the center of the sunspot for detected objects were also computed.

3.4.3 Computing the line profile

The “master” average line-profile for PMJs in the Ca ii 8542 Å  line was computed using the line-profiles of the center-of-mass-pixels and their 8 neighbour-pixels for each individual detection. The average was thus carried out for only those PMJ detection-areas through all timeframes that actually contained the center-of-mass-pixel, as computed from the detection area for each detection, as well as this pixel’s 8 pixel-neighbours. This selection was performed to ensure that only line-profiles of pixel-positions that did not fall outside of the actual PMJ detection areas (as could be the case for some center-of-mass-pixels due to for example peanut-shaped PMJ areas), and to make sure only pixel-positions that represented well-formed PMJs were included. The final average line profile is thus based on a subset of the 4253 PMJ detections through all timeframes totaling 3953 PMJ detections, as not all PMJ areas contained their theoretically computed center-of-mass pixel, or because the area did not contain all 8 immediate pixel neighbours of the center-of-mass pixel.

For the computation of the average line-profile a total of $3953\cdot 9=35577$ pixel-position line profiles was therefore used. The entirety of pixel positions of all PMJ detection areas were not employed for simplicity, and as not to skew the line profiles towards larger PMJs, which would then have contributed more strongly towards the profile (though a difference in line-profiles between large and small PMJs has not been investigated).

Finally, the individual 9-pixel-average line-profiles of each of the PMJs in the mentioned subset were also investigated with regards to distinct peaks in the blue and red wing as well as their line-core minimums. The wavelength positions of the peaks and the minimum and their final average values were estimated by interpolating the individual profiles, and then using a sliding window approach to find the local peaks and minimum in each given profile. Profiles for the average of different categories of profiles (profiles with blue and/or red peaks present, profiles with both blue and red peaks present, profiles with just either of the peaks present) were also computed to compare to the main average and each other. The resulting values for the positions of the average minimum, blue and red peaks were calculated by averaging the found positions in the individual interpolated line profiles, and are thus not limited to the 37-sampling points (but are therefore subject to greater uncertainty).

4 Results

The automatic detection of PMJs as outlined in the last section made it possible to collect a large and statistically significant set of PMJ detections with associated properties that could then be investigated.

In the sections below the different derived properties and statistical values associated with these events are presented.

4.1 Detection summary

Table 1 summarizes the detections performed by the automatic detection pipeline, together with the most basic statistics derived from them. It is evident that the automatic approach of identifying PMJs using their spectral profile yielded a large dataset from which to infer PMJ properties. With a number of PMJs equaling 453 tracked events, derived statistical values will be significant. Each individual frame in the detections contains an average of approximately 21 PMJs, and this rate highlights the continuous occurrence of PMJs throughout the observations. As each of these detected PMJs is tracked through time, but with unique associated properties in each timeframe, the number of individual PMJs summed over all timeframes is greater than the 453 tracked events. This number is 4253 PMJs present throughout all individual frames and corresponds to the 453 distinct, tracked events (see table 1). However, some further selection is performed before some statistics were computed (namely line-profile averaging as well as intensity and line-profile feature investigations, see Section 3.4.3).

Figure 4 depicts a sample frame from the observations at an offset of $-275$ mÅ in the Ca ii 8542 Å  line with PMJ detections shown. Visible in the figure are the overlain PMJ detection-areas, as well as the associated computed ellipses which were used to measure the lengths and widths of the PMJ detections.

As a further reasonability check, a check of the relative brightness in the detection areas of all PMJs was performed. Here, the individual average brightness in both the Ca ii 8542 Å line at an offset of $-275$ mÅ and in the Ca ii H line core was computed for each PMJ detection area. These were normalized to the average brightnesses for each individual frame of the penumbral region (as outlined in Fig. 1) in each of the two passbands. PMJs are seen as bright features in by-eye detections in both wavelengths, and thus one would expect an average brightening in the detection areas for both. Furthermore, as the detections are carried out ultimately utilizing the Ca ii 8542 Å line profile, an average brightening in the detection areas in the Ca ii H line core would further strengthen the co-occurrence of PMJs in both passbands. It was found that $80\%$ of PMJs had an average relative brightening above unity in the Ca ii H line core and that $79\%$ of PMJs had an average relative brightening above unity in the Ca ii 8542 Å line at an offset of $-275$ mÅ. Furthermore, $71\%$ of detections show a simultaneous brightening above unity in both passbands throughout the times series. These values strengthen the assumption that bright features are being detected in the brightness-independent detection scheme utilizing the Ca ii 8542 Å line profile shape. As the brightness of the PMJs was estimated as an average of their entire detection area, their brightness as compared to the average of the entire penumbra for a given timeframe may in fact be a conservative comparison. A comparison between for example the brightest pixel and the penumbra average for any given PMJ detection would likely yield a higher relative brightness in both passbands. This is due to automatically detected PMJs tending to exhibit a larger area than if selected solely by-eye compared to the surrounding intensity. Comparing PMJ brightnesses to the local average brightness of the penumbra may very well have resulted in a higher relative brightnesses as well.

4.2 Basic PMJ statistics and properties

Having computed the length and width for all PMJ detections using the minor and major axis of the fitted ellipses (as described in Sec. 3.4.1), they could be presented as distributions. Figure 5 gives these histogram distributions for the lengths and widths of all detected PMJs throughout the observations.

The distributions for lengths and widths seem to be well behaved with trailing ends tending towards zero, as would be expected. The lower ends of the distributions also seem to taper off in the last lower length/width bins. One pixel in the observations corresponds to 43 km, thus both distributions’ lowest length/width bins with entries include sizes larger than this, however the detections were limited by a lower-limit area, which will also effect the lower range. The average width/lengths must thus be interpreted with this in mind. The associated mean length of $640$ km and mean width of $210$ km (see also Table 1) are reasonable, and are consistent in rough magnitude with values reported in Katsukawa et al. 2007, namely lengths of $1000-4000$ km and widths of approximately $400$ km or less for the PMJs in Hinode Ca ii H. These values are also consistent with the mega meter range as given in Reardon et al. 2013 when observing individual PMJs (or penumbral transients) in the Ca ii 8542 Å  line. Both estimates found here are however still considerably smaller, almost half for both values.

In Fig. 6 the distributions of durations of tracked PMJs is given with a cut-off value of 8 min, representing $96\%$ of tracked detections, not showing outliers.

These outliers were excluded because they for the most part represented a small fraction of tracked PMJs that had extremely long lifetimes. The truncated distribution represents $96\%$ of tracked PMJs. The abnormally long lifetimes were most likely due to detections in areas in which PMJ detections were ubiquitous throughout the observations, and where PMJ events overlapped so closely in time and space that they were tracked continuously as one event. Also, spurious and long lasting detections may also have been caused by the strong inverse Evershed flow (see Fig. 1) present on the center side of the observed sunspot.

4.3 Ca ii 8542 Å PMJ line profile

The PMJ line-profile in the Ca ii 8542 Å  line computed from the average of the $3953\cdot 9=35577$ pixel-positions is given in Fig. 7 together with reference profiles for the quiet Sun and the penumbra. The PMJ profile is characterized by enhanced inner wings (at about $\pm 385$ mÅ and a brighter line core (with the core being at 116% of the quiet sun line core brightness). The enhancement of the inner blue wing is stronger than in the red wing. The far wings of the PMJ profile approach the level of the average penumbra. This average profile is less pronounced than individual profiles of PMJs selected by eye and inspecting individual pixel-position profiles (which will be made more evident in section 4.4).

The found average PMJ profile is however still consistent with other reported profiles as given in Reardon et al. 2013 (their Fig.4) and Vissers et al. 2015 (their Fig.5), but does show a less pronounced peak in the blue, and a very weak enhancement in the red compared to most of these published profiles, but is still recognizable. It must be emphasized that the profile presented here is an average, computed from many individual profiles, whilst the profiles it has been compared to in previous work are profiles from individual pixel positions of by-eye-selected PMJs. These individual profiles were most likely selected specifically due to their distinct features, meaning that the average profile presented consequently will not present as sharp features. Individual profiles of PMJs in the presented observations (both selected by eye, as well as those contained in PMJ detection areas) still exhibit such distinct features to a large degree, as will also be exemplified in Sec. 4.4.

As mentioned in section 3.4.3, when averaging the master line profile, each individual (9-pixel average) profile corresponding to individual detections was also inspected for the presence of a tell-tale blue and/or red peak in the line. As a result, it was possible to create averages of the profiles according to the presence of peaks in the blue or red. Thus, averages for the profiles for the cases of both a blue and red peak being present, one or the other being present or just one of the peaks being present could be computed. Figure 8 shows these average Ca ii 8542 Å line profiles for different subsets of PMJ detections.

For completeness, Table 2 summarizes these different groups of profiles. Note especially that only less than half (N${}_{\text{profiles with peaks}}=1868$) of all 9-pixel average PMJ profiles used in the peak detection (N${}_{\text{all}}=3953$) have clear automatically detectable peaks. Further, a vast majority of these were in the blue, with a total number of blue peaks of N${}_{\text{blue peaks}}=1725$ and a total of red peaks of N${}_{\text{red peaks}}=143$. This corresponds to percentages of $92\%$ and $8\%$, of blue and red peaks respectively, of the total number of detected peaks. Further, of all investigated averaged 9-pixel profiles, $43\%$ exhibit blue peaks whereas only $3.6\%$ exhibit detectable red peaks.

From Fig. 8 it is evident that the average line profile of those profiles that have either both or one peak present (magenta-dashed-dotted in the figure) is near identical to the average line-profile for the profiles that have blue peaks (and no red peaks) present (blue-dashed in the figure). This is concurrent with the fact that there is a much larger number of profiles with blue peaks compared to red peaks. The average line-profile of profiles with only red peaks is correspondingly significantly different from both these profiles and only shows a peak in the red (note however the low number of 29 profiles in this average). All three profiles are more distinct in appearance than the overall average line-profile of all used (9-pixel) line profiles (solid-black in the figure).

Finally, and most notably, the average line-profile of the profiles with both blue and red peaks present simultaneously, which therefore has well-defined peaks in both the blue and red, more strongly resembles the profiles of by-eye selected PMJs in the present dataset (which again, will be made more evident in section 4.4). For likely the same reason, this profile also more strongly resembles reported profiles as found for by-eye detected PMJs in the Ca ii 8542 Å line given in Reardon et al. 2013 (Fig.4) as well as several of the distinct profiles presented in Vissers et al. 2015 (Fig.5). This profile is also of overall greater intensity in both the peaks as well as in the line-core minimum compared to the overall PMJ average profile and the FOV average of the Ca ii 8542 Å line. In fact, the line core is at an 145% intensity compared to the quiet sun line core average, a clearly greater enhancement than for the overall line profile average, which as given earlier exhibited an intensity of only 116% compared to the quiet sun. This makes it plausible that the higher overall intensity in these types of PMJs makes it easier to pick them out by eye, and thus makes the presence of both peaks in these selections more likely, whereas the automatic detection presented may not be as susceptible to this bias (after the initial selection of the k-NN reference set). An over-selection of PMJs with strong enhancements in the red wing of the Ca ii 8542 Å line may therefore be likely in by-eye detections.

It is worth noting that the average PMJ profile of the kNN reference set shown in Fig. 3 exhibits a similar shape to the average profile of detected PMJs with both blue and red peaks present, and thus has more clear peaks than the final PMJ average profile of all PMJ detections. It also has an overall higher intensity than the total average, though not as high as the average with both blue and red peaks present. This similarity is most likely due to the previous point raised that by-eye selection favours PMJs with strong enhancements in both the blue and the red, as well as overall brighter PMJs.

We investigate the spatial distribution of the PMJ profiles with distinct blue and red peaks. It may be well plausible that the average PMJ orientation is related to the large scale magnetic field topology of the sunspot so that the inclined viewing angle may have an effect on the spatial distribution of the observed PMJ profiles. The results are presented in distribution graphs in polar coordinates, centered on the umbra and the limb direction approximately $90\degree$.

In Fig. 9, a scatter plot of the peak-intensities versus the angle around the center of the sunspot of the automatically detected peaks in the red and blue of the Ca ii 8542 Å line for the detected PMJs is given. There is no clear discernible bias in the plot with regard to the intensity, both for the blue and red peaks. However, there is a readily apparent clustering of red peaks in the degree range of $315-30$, with two noticeable groups within this range. The group above the zero-degree mark seems to neatly coincide with the principal PMJ hot-spot as described in Sec. 4.4 below (see Fig. 11). The presence of more readily detected red peaks in an area in which many PMJs are found throughout the timeseries is intuitive, as a high count of PMJs should lead to a higher count of red peaks as well. On the other hand, the area still seems overrepresented in the amount of red peaks compared to other areas with significant numbers of PMJs, and may indicate that red peaks are favored in some areas.

In Fig. 10, a scatter plot of the wavelength-positions versus the angle around the center of the sunspot of the automatically detected peaks in the red and blue of the Ca ii 8542 Å line for the PMJ detections is given. Here, there is an apparent bias in the degree of red shift in some of the red peaks, whereas there is no clear bias in the blue shift of the blue peaks. We can again discern the general clustering of red peaks in the same range as for Fig. 9, but this time there is a clear difference in the two groups within this angular range. The group coinciding with the PMJ hot-spot (see again Sec. 4.4 and Fig. 11) at slightly below 30 degrees has a clearly lower red-shift, and is less spread out in values, than the group of red peaks clustered around 330 degrees (see also the reference line at $10$ km s*-1* in the Figure). The atypical red shift of the red peaks situated around the 330 degree mark are likely largely caused by a strong inverse Evershed flow that moves into the penumbra at this location, which may also account for the greater overall spread of red shifts in this region. This region of strong inverse Evershed flow is an extension of the large dark cloud in the bottom middle part of the right panel of Fig. 1. We observe these inverse Evershed “clouds” move into this part of the penumbra throughout the full duration of the time series. A detailed inspection of the spectral profiles with CRISPEX of this region reveals that at times, the profiles are largely affected by this inverse Evershed flow.

There also seems to be an overall higher blueshift in the detected blue peaks of the Ca ii 8542 Å line profiles on the limb-side of the Penumbra (in the 30 to 180 degree range), which is counter intuitive to a line-of-sight enhancement of the blueshift, as this would be expected for the disk-side instead. This will be mentioned further in Section 5.

4.4 Clustering and a near-continuous occurrence of PMJs

Figure 11 depicts a density map of all the individual PMJ detection-pixels, summed over all timeframes. PMJs are detected rather evenly distributed over the azimuthal direction, and mostly in the outer half of the penumbra.

From the density distribution it is evident that there are preferred sites for PMJ formation and that they are not evenly distributed throughout the sunspot’s penumbra. We can see a clear clustering of PMJ detections in certain regions, and two distinct regions in the upper right corner in particular. These regions are the sites of a large number of PMJs throughout the observations.

There is an apparent bias in the number of detections with regards to position in the penumbra, as it is readily seen that there are many more detections on the limb-side of the sunspot. The right-upper corner of the penumbra mostly coincides with the far-side of the observer’s line-of-sight, which might possibly contribute to a larger number of clear detections. Reciprocally, the lower side of he sunspot may exhibit a lesser amount of detections due to foreshortening of the (nominally) elongated PMJs. There are however still distinct hotspots of PMJ-activity on the same side of the sunspot, meaning that a clustering of PMJs could not be caused by foreshortening effects to a large degree.

Figure 12 highlights the behavior of a “PMJ-hotspot” (a site of repeated PMJ activity) picked out by-eye using CRISPEX (see legend for details). The timeslices in particular, as well as the intensity-time plots, highlight how the PMJ-events seem to present as continuous processes of waxing and waning in intensity in both the Ca ii 8542 Å  and the Ca ii H line, as opposed to very well-defined one-off events with clear onsets and ends. Both events highlighted in these figures are situated in a PMJ-hotspot as readily seen in Fig. 11. We can observe the typical behavior of PMJs in the present observations, namely being localized to specific regions, and to seemingly reoccur over time. This ongoing process of PMJ generation at preferred sites may possibly be due to favorable magnetic field structures at these particular sites, leading to repeated magnetic reconnection in quick succession.

Figure 13 highlights the Ca ii 8542 Å  line profiles of three specific pixel positions, all contained within a single PMJ detection area, as shown in Fig. 4. Shown in Fig. 13 are the profiles for a pixel position on the umbra-side of the detection area, the center-of-mass pixel position of the area, and a pixel position on the outer side of the area. These line profile examples highlight that the automatic detection envelops a larger area than what would perhaps be picked out as a singular PMJ in by-eye detections (see Fig. 4), but that the contained profiles still have the distinct PMJ shape. In particular, this event may be categorized as two distinct jets using by-eye detections at the shown wavelength offset, whereas the automated method based on the line profiles identifies the entire area as one PMJ. The three different profiles highlighted all exhibit the distinct PMJ peak in the blue. There is also a clear enhancement in the red, but this is most evident for the center-of-mass position and the reference point closest to quiet Sun. The reference profile for the umbra-side position has some enhancement in the red, but is less clear. The two profiles closest to the umbra are very much distinct PMJ profiles, whereas the center-side profile is generally more subdued, but still with enhancement in the blue and red peak positions. The profile also has as a clearly enhanced line core compared to the penumbra average line profile. These profiles are generally very similar to by-eye selected profiles (as for example seen in Fig. 12). The profiles in Fig. 13 also exhibit an incremental increase in intensity for the blue and red peak wavelength offsets as one moves towards the quiet sun for the overall profiles. This trend is not present for the line core however, with the line core intensities seeming fairly stable.

5 Discussion

5.1 Similarity to Ellerman bombs

The typical PMJ Ca ii 8542 spectral profile with enhanced inner wings resemble the characteristic spectral profile of Ellerman bombs, a similarity that was already pointed out by Reardon et al. 2013 and Vissers et al. 2015. Ellerman bombs (Ellerman 1917) are the telltale signature of magnetic reconnection in the low atmosphere, usually associated with emergence of strong magnetic flux in active regions (see Rutten et al. 2013 for a recent review). Like for Ellerman bombs, magnetic reconnection is the driving mechanism for PMJs that is favored in the literature (see e.g. Katsukawa et al. 2007, Ryutova et al. 2008 and Nakamura et al. 2012 ). Ellerman bombs are commonly observed in H$\alpha$ where they exhibit enhanced wings but an undisturbed line core which is an indicator that the reconnection occurs below the chromospheric canopy (see, e.g., Watanabe et al. 2011). Vissers et al. 2013 studied Ellerman bombs in H$\alpha$ and Ca ii 8542 from a CRISP data set that was acquired just before the data we present here. The Ca ii 8542 Ellerman bomb profile has enhanced wings like in H$\alpha$ (see their Fig. 5) and a dark line core from obscuration by overlying chromospheric fibrils.

It is worth to emphasize that the wing enhancement in PMJs is much more modest than in Ellerman bombs where the intensity level in the inner wing rises above the far wing quiet Sun level and reaches far into the wings, beyond 1 Å offset from the line core. In PMJs the enhancement extents mostly to about 0.6 Å and stays well below the far wing quiet Sun reference intensity. Further, we note a 116% intensity of the Ca ii 8542 average PMJ line core as compared to the quiet Sun (and a 145% intensity for the line profile average with both blue and red peaks present). In the line core images, PMJs are not obviously obscured by overarching fibrils like for Ellerman bombs, and sometimes we can discern bright features that resemble the PMJ morphology at $-275$ mÅ offset, see the bottom panel of Fig. 2 for an example. We regard the enhanced line core as consistent with the observation that PMJs get heated to transition region temperatures as reported by Vissers et al. 2015 who found signs of progressive heating along the PMJ length from coordinated IRIS and SST observations.

5.2 Spatial distributions

Besides the enhancement of both inner wings in PMJ line profiles, we note that there is a preference for larger enhancement of the blue wing. For those profiles where the enhancement is in the form of one or two clearly identifiable inner wing peaks (close to half of the detected PMJs), the majority has a blue peak. This is also reflected in the average PMJ profile that displays a clear blue-over-red asymmetry. We investigated whether there is any trend in the spatial distribution of the properties of the inner wing peaks. If one would (perhaps naively) interpret PMJs as near-vertical plasma upflows, one would expect this to have an imprint on the observed spectral profiles for a sunspot under this observing angle (observing angle $\theta=57\degr$). This viewing angle effect is for example very clear for the photospheric Evershed flow in sunspots, where Dopplermaps show a clear, highly inclined, outflow in the form of red-shifts at the limb-side and blue-shifts at the center-side for sunspots away from disc center (see, e.g., Scharmer et al. 2011, for a recent example) We find, however, no clear systematic imprint on the spatial distribution of spectral parameters of PMJ profiles. The peak intensity of the inner wing peaks show no trend in the spatial distribution over the sunspot (see Fig. 9). There may be a trend of higher blue-shifts of the blue peaks at the limb-side penumbra (Fig. 10) where the blue peaks are all shifted more than $-10.5$ km s*-1*. On the center-side, the shifts are between $-7$ and $-14$ km s*-1*. This trend goes against the simple interpretation of the shift of the blue peak as a pure Dopplershift from a upflow in the expanding chromospheric magnetic field of the sunspot – in such scenario the strongest blueshifts would be found in the center-side penumbra. We conclude that detailed numerical modeling with a realistic treatment of the radiative transfer in the optically thick penumbral atmosphere is required in order to interpret the PMJ spectral profiles. A complicating factor is the inverse Evershed effect in the form of (mostly) red-shifted clouds that at certain times and spatial locations result in strongly affected line profiles. Part of the profiles that were identified as “red-peak” PMJ profiles were clearly affected by inverse Evershed clouds that were unrelated to PMJs.

5.3 Spatial dimensions

From our large statistical sample of automated PMJ detections, we determine PMJ lengths that are on the short side as compared to the measurements from Hinode Ca H observations (Katsukawa et al. 2007). To large extent this can be attributed to our method’s sensitivity to weaker events and the inclusion of the center side penumbra, where the projected PMJ extensions suffer from foreshortening. The measurements from previous studies have an intrinsic bias towards longer PMJs from by-eye selection.

5.4 Lifetimes

We determine an average lifetime of 90 s which is longer as compared to the typical lifetime of less than 1 min reported earlier (Katsukawa et al. 2007). However, we note that we find a large number of short-duration events (the median lifetime is 75 s). We have discarded long-duration detections ($>$8 min) that were in part resulting from clustering of individual PMJs occurring in close vicinity and rapid succession, and in part due the earlier described strong inverse Evershed flow on the disk side of the sunspot, distorting the Ca ii 8542 Å line profile, causing false identifications. Thus, we cannot exclude the possibility that intermediate duration detections (3-8 min) are also affected by neighbouring PMJ activity. However, we decided to be conservative in manually sifting though the detection statistics. Further, we note again our method’s sensitivity to weaker events which allow us to track events for longer duration as compared to manual and by-eye selection methods.

6 Conclusions and Summary

We studied PMJs using an automated simple machine learning detection scheme consisting of an initial Principle Component Analysis for the compression of data, and the subsequent application of the k-Nearest Neighbour algorithm and finally simple object tracking over the timeseries. This scheme was applied to high spatial resolution observations of well-sampled Ca ii 8542 profiles. We verified that the automated detections of PMJs in Ca ii 8542 match well with PMJs in co-temporal Ca ii H linecore filtergrams, the diagnostic for PMJs used in earlier studies. The Ca ii 8542 PMJ line profile is characterized by enhanced inner wings, often in the form of clear peaks, preferably with a distinct asymmetry towards stronger blue wing enhancement. The line core is enhanced as compared to the quiet Sun reference spectrum. We detect a total of 4253 PMJs at a detection rate of 21 events per timestep over a duration of 41 min corresponding to 453 PMJs tracked in time. Ellipse fitting to the PMJ detection areas yield average PMJ dimensions of 640 km length and 210 km width. We measure an average lifetime of 90 s (discarding the longest duration events, $>$8 min, that are clearly separate but overlapping events in rapid succession or the result of misidentifications). We detected PMJs in all parts of the penumbra, with many detections on both the limb-side as well as on the disk-center-side of the penumbra. However, there was still an apparent bias in that there were more detections on the limb- or upper- side of the sunspot, perhaps in part caused by foreshortening effects. We note the existence of clear “hot-spots” with high occurrence rates of PMJs.

We finally remark that our results contribute to a solid observational characterization of PMJs which is needed as constraints for theoretical and numerical modeling. Further research will necessarily have to focus on numerical studies to elucidate the precise physical nature of PMJs. Quantification of the heat-energy transfer by PMJs into the higher sunspot atmosphere is one such area of interest for future investigations.

Acknowledgements.

The Swedish 1-m Solar Telescope is operated on the island of La Palma by the Institute for Solar Physics of Stockholm University in the Spanish Observatorio del Roque de los Muchachos of the Instituto de Astrofísica de Canarias. Our research has been partially funded by the Norwegian Research Council and by the ERC under the European Union’s Seventh Framework Programme (FP7/2007-2013) / ERC grant agreement nr. 291058. We made much use of NASA’s Astrophysics Data System Bibliographic Services.