Investigation on the Spatiotemporal Structures of Supra-Arcade Spikes

We carried out a survey of flares with magnitude of M-class and above in the SDO era to search for events in which the post-flare arcade is viewed side-on so that the supra-arcade spikes (SASs) are visible. To make certain that a magnetic flux rope is involved in the eruption, we required that the associated CME eventually arrives at Earth and is observed as a magnetic cloud (Burlaga et al., 1981; Burlaga, 1988) by a near-Earth spacecraft. However, such events are relatively rare, because SASs are preferentially visible close to the limb, while CMEs originating from the disk center are more probable to arrive at Earth than those from near the limb. Eventually we found four events as listed in Table 1; each flare is associated with a halo CME that is observed in situ as an ICME 3–4 days later.

The SASs are preferentially observed in 131 Å (Fe \@slowromancapxxi@ and Fe \@slowromancapxxiii@), fairly visible in 94 Å (Fe \@slowromancapxviii@) due to inferior signal-to-noise ratio and cooler characteristic temperature in this passband, and often invisible in other even “cooler” passbands of the Atmospheric Imaging Assembly (AIA; Lemen et al., 2012, see Figure 1) on board the Solar Dynamics Observatory (SDO; Pesnell et al., 2012). Hence we focuse on the 131 Å passband in this study. The AIA takes full-disk images with a spatial scale of $0^{\prime\prime}.6$ pixel^-1 and a cadence of 12 s. In particular, Fe \@slowromancapxxi@ dominates the 131 Å passband for the flare plasma, with a peak response temperature $\log T=7.05$, and Fe \@slowromancapviii@ dominates for the transition region and quiet corona, with a peak response temperature $\log T=5.6$ (O’Dwyer et al., 2010). Previous studies applying the differential emssion measure technique to AIA’s six optically thin passbands, i.e., 171, 193, 211, 335, 94, 131 Å (cf. Figure 1) generally demonstrated a significant presence of $\sim\,$10 MK plasma above the post-flare arcade (e.g., Gou et al., 2015). Table 1 lists publications that study the supra-arcade plasma of the same flares.

The CMEs are observed by the C2 and C3 coronagraphs of the Large Angle and Spectrometric Coronagraph Experiment (LASCO; Brueckner et al., 1995) on board the Solar and Heliospheric Observatory (SOHO; Domingo et al., 1995). The CME speed in the inner heliosphere (within 30 solar radii) is estimated through a linear fitting to the height-time measurements of the CME front and recorded by the SOHO LASCO CME Catalog²²2https://cdaw.gsfc.nasa.gov/CME_list/. The ICMEs are detected in situ as they traverse the Wind spacecraft near the Earth. We mainly consult the ICME list³³3http://space.ustc.edu.cn/dreams/wind_icmes/ compiled by Chi et al. (2016). For each ICME, there could be multiple CME candidates within a time window of 36 hrs, i.e., about 3–4.5 days before the ICME arrival at the Earth, roughly corresponding to an average propagation speed of 400–600  $\mathrm{km}\,\mathrm{s}^{-1}$ between the Sun and Earth. We determine the source of the ICMEby carefully taking into account the CME characteristics such as its location, direction, speed, size, timing, etc. The CME-ICME association for Event #2 has been well established by Vemareddy & Mishra (2015).

Figures 2–5 show the spikes detected in a typical timescale of 2 hrs during the flare gradual phase in each of the four events, when the spikes are clearly observed in the AIA 131 Å passband. To detect the spikes, we placed a virtual slit above the post-flare arcade. The slit is either a straight line or a hand-drawn curve to follow the shape of the post-flare arcade (see panel (a) of each figure). Since the brightness along spikes diminishes with height quickly, the slit is placed not far away from the post-flare arcade. The brightness along the slit is shown in panel (b) of each figure. A visual inspection indicates that each major peak in this plot corresponds to a spike in the image. Note, however, that these major peaks are superposed by rapid wiggles that are comparable to the measurement errors given by AIA_BP_ESTIMATE_ERROR (shown as error bars). To identify the major peaks but discard the wiggles, we first smooth the brightness profile along the slit by a five-point running average (shown as the red curve in (b)), and then locate the local maximums and minimums in a running box of 13 pixels. All local maximums must exceed 20 DN pixel^-1 s^-1, an empirical background set in this study. The distance between two nearest local minimums found at each side of a spike is taken as its width (cf. §3.1). The identified major peaks are marked by red arrows on the top of panel (b).

Further, for the two disk events, #2 and #4, AIA 131 Å images are co-registered to the 1st image taken during each selected time interval using the SSW procedure DROT_MAP. We made the time-distance stack plot (panel (c)) by taking a slice along the virtual slit from each image and then arranging the slices in chronological order. The temporal evolution of spikes is then recorded in this time-distance diagram. One can see that some spikes evolve consistently, some are wavering due to SADs, and others may apparently appear or disappear during the investigated time period. It is interesting that some SADs reveal themselves as dark, tadpole-like features in the time-distance diagram as they propagate through the slit (e.g., Figure 2c). It can be seen that SADs may flow not only along intra-spike “channels” but also into the spikes, the latter of which may result in the apparent splitting of a spike (some clear examples can be seen in Figure 3c). Such interactions between bright spikes and dark down-flowing voids are suggested to be evidence for Rayleigh–Taylor type instabilities in the exhaust of reconnection jets (Guo et al., 2014; Innes et al., 2014).

The identified spikes along the slit are plotted as crosses in Panel (d), which has the same time and distance axes as the time-distance diagram in Panel (c). We find our identification algorithm to be satisfying though a visual comparison between (c) and (d). The time-averaged number of spikes and the standard deviation are annotated on the top right of panel (d).

With the width of each spike being estimated (§2.2), we plot the distribution of spike widths during the gradual phase of each flare (Figure 6). The distribution of all four flares investigated is rather similar, with a peak at around 5–6 Mm, an average of 16–21 Mm, a standard deviation of 7–11 Mm, and a median of 15–19 Mm. The peak of 5–6 Mm is well consistent with the current-sheet width detected in the lower corona in EUV passbands as reported in the literature (e.g., Liu et al., 2010; Zhu et al., 2016). Each distribution also has a similar lower cutoff of about 3 Mm and an upper limit of about 20 Mm. The lower cutoff can be attributed to the running box of 13 pixels specified in our algorithm, which effectively limits the spike width to be above 6 pixels, or 2.6 Mm. This is smaller than the lower limit of the current-sheet width, 3 Mm, deduced by Guo et al. (2013) from the nonlinear scaling law of the plasmoid instability. Note Lin et al. (2007, 2009) gave a larger lower limit (50 Mm) based on the linear scaling law.

Apparently, spike widths are not normally but log-normally distributed. In Figure 6, employing a nonlinear least squares fitting procedure MPFIT⁴⁴4http://purl.com/net/mpfit (Markwardt, 2009) and assuming a 2-pixel measurement error, we fit each distribution with the following function,

where $C$ is an arbitrary constant. The other two parameters $(\mu,\sigma)$ are annotated in the middle right of each panel. It is noteworthy that we obtain similar $\mu$ and $\sigma$ in all four events.

We apply the Fourier transform to each vertical slice in the time-distance diagram in Panel (c) of Figures 2–5 to explore the structure of the supra-arcade emission in terms of spatial frequency $k$ (Mm^-1). We fit the Fourier power spectrum resulting from each vertical slice with a power-law function and show the temporal variation of the power-law index $\gamma_{s}$ in the left column of Figure 7. Collapsing the time-distance diagram along the time axis gives the time-integrated EUV emission profile along the slit. Applying the Fourier transform to this profile, we obtain power spectra exhibiting clear power-law behaviors (right column of Figure 7), whose fitted indices are in the range of $\sim\,$[4, 5], similar to the range of $\gamma_{s}$ in the left column.

Similarly, we apply the Fourier transform to each horizontal slice in the time-distance diagram in Panel (c) of Figures 2–5 to explore the structure of the supra-arcade emission in terms of temporal frequency $\nu$ (mHz). Fitting the Fourier power spectrum from each horizontal slice by a power-law function, we show the variation of the resultant power-law index $\gamma_{t}$ along the virtual slit in the left column of Figure 8. Collapsing the time-distance diagram along the distance axis, we effectively integrate the EUV emission along the slit, which emulates the situation when the VCS is observed edge-on in the optically thin corona. This results in a time series of EUV emission sampled from a point on the vertical current sheet. The power spectra of spatially integrated EUV emission also exhibit clear power-law shapes (right column of Figure 8), whose indices are in the range of $\sim\,$[2.5, 4], similar to the range of $\gamma_{t}$ in the left column.

Cheng et al. (2018) construct time-distance diagrams from a slit along a VCS observed edge on. Sampling two horizontal slices above the flare arcade in a time-distance diagram, they obtained the corresponding Fourier power spectra with a power-law indices $\alpha$ of 1.8 and 1.2, respectively. The difference in power-law indices between $\gamma_{t}$ in our case and $\alpha$ in Cheng et al. (2018) can probably be attributed to the following differences: 1) the pow spectra in Cheng et al. (2018) appear very noisy (their Figure 13), lacking a strict power-law shape, compared with the nice power-law behavior demonstrated in Figures 7 and 8; and 2) the line-of-sight integration in their case includes the coronal foreground and background, whereas in our case the spatial integration is limited to coronal emission right above the post-flare arcade, therefore less interferences from irrelevant coronal structures.

Below we demonstrate heuristically the consistency of the observed power-law behavior with Kolmogorov turbulence in the supra-arcade region. Any turbulent physical quantity can be written as

where $u_{0}(x,t)$ varies smoothly at large spatiotemporal scales, while $\delta u(x,t)$ fluctuates randomly at small scales, i.e., there is a large separation of both space and time scales between the “mean” and fluctuating components. The two components hence respond distinctively to a long wavelength average, i.e.,

The Fourier transform of the observed intensity $I=I_{0}+\delta I$ gives the power spectra,

in terms of spatial frequency $k$ and temporal frequency $\nu$, respectively, as shown in Figures 7 and 8. Here for clarity, the spectral index $\gamma_{s}$ and $\gamma_{t}$ are set as positive. $I_{0}$ is transformed to the low-frequency end of the power spectra, and $\delta I$ to the higher-frequency portion with power-law behavior, i.e., $\delta I^{2}\sim E(k)k\sim k^{-\gamma_{s}+1}\sim\mathnormal{l}^{\,\gamma_{s}-1}$, which gives

because $\gamma_{s}$ is found to be in the range of [4, 5]. Similarly, since $\delta I^{2}\sim E(\nu)\nu$, we have

with $\gamma_{t}$ in the range of [2.5, 4].

On the other hand, the intensity of coronal emission lines $I\sim\int n^{2}ds$ (Aschwanden, 2005, §2.8), where $s$ is the distance along the LOS. Expanding $n$ according to Eq. 2, we have

The LOS integration a priori takes a long wavelength average, which gives

As a comparison, we anticipate that the turbulent component would be smoothened out in white-light observations of the K-corona, in which the observed intensity is dominated by Thomson scattering, i.e., $I\sim\int nds$.

The density perturbation $\delta n$ is linked to the velocity fluctuation $\delta\mathbf{u}$ by the equation of continuation

assuming that $|\delta n|\ll n_{0}$ but $|\partial n_{0}/\partial t|\ll|\partial\delta n/\partial t|$ and $\mathbf{u}_{0}\simeq 0$. Thus,

in orders of magnitude, where $t_{0}$ and $L_{0}$ are characteristic time and length scale of the system, respectively. The assumption that the Alfvén speed $V_{A}$ is on the same order of magnitude as $L_{0}/t_{0}$ implies that $|\delta u|\ll V_{A}$. This is consistent with spectroscopic observations of hot emission lines in the VCS, which give nonthermal velocities of no larger than 200  $\mathrm{km}\,\mathrm{s}^{-1}$ (e.g., Li et al., 2018). We further argue that the LOS integration from the side-on perspective of the VCS make relevant the size $l$ of large eddies in the flow velocity field $\mathbf{u}\simeq\delta\mathbf{u}$, with the thickness of the VCS being comparable to those of spikes; i.e., $\delta I\sim\delta u^{2}\,l$, with $l\lesssim L_{0}$. $l$ can be deemed as the order of magnitude of distances over which $\delta u$ changes appreciably.

If the observed intensity perturbations $\delta I$ result from a fully developed turbulence following the famous universal law of Kolmogorov and Obukhov, i.e., $\delta u\sim\mathnormal{l}^{1/3}$ (Landau & Lifshitz, 1987, §33), then we have

The index 5/3 falls nicely in the compact range of [3/2, 2] as given independently by Eq. 5. With $\nu\sim(l\,\delta u)^{-1}\sim l^{-4/3}$, we have

in which the index also falls in the compact range of [3/4, 3/2] as given by Eq. 6.

Thus, when the fluctuations in coronal emission-line intensity observations are dominated by turbulence, the corresponding power spectra are expected to exhibit the power-law behavior as follows,

As a further evidence of turbulence in the VCS region, one can see from the left column of Figure 8 that the power-law index $\gamma_{t}$ is generally closer to the value of 7/2 in the middle of the slit, that is, within the VCS region, than at the two ends of the slit, that is, outside of the VCS.

In contrast, the LOS integration from the edge-on perspective of the VCS gives $\delta I\sim\delta u^{2}\,L$, with the ‘depth’ of the VCS being comparable to the axial length of the post-flare arcade, i.e., $L\gg L_{0}$. This leads to power spectra in $k^{-7/3}$ or $\nu^{-2}$, which are only slightly steeper than those obtained by Cheng et al. (2018).

Each SAS can be considered as a reconnection site on the 3D VCS in the wake of the CME. According to the standard model, each episode of magnetic reconnection of the overlying field at the VCS adds a twist of roughly 1 turn to the burgeoning CME flux rope. If magnetic reconnections at each SAS proceeds continually, i.e., there is no significant time interval between any two episodes of magnetic reconnection on the same SAS, then at any time one may assume that the number of ongoing reconnection episodes roughly correspond to the number of SASs. Since the number of SASs is stable during the flare decay phase, we expect that the flux rope’s outer shells are more or less uniformly twisted (e.g., Wang et al., 2017). When the CME arrives at the Earth, one would expect near-Earth spacecrafts to detect a magnetic cloud possessing magnetic twist of similar number of turns in its outer shells as the number of SASs.

We fit the four ICMEs by a velocity-modified uniform-twist force-free flux rope model (Wang et al., 2016). The in-situ observations and corresponding fitting results are shown in the Appendix Figures 11–14. The model yields the twist density $\tau=rB_{\phi}/B_{z}$ in local cylindrical coordinates $(r,\phi,z)$ with the $z$-axis directed along the flux-rope axis. $\tau$ is given in units of the number of field-line turns per unit AU along the flux-rope axis. The helicity (twist) signs are negative for all four events, which, except Event #3, originate from the northern hemisphere, consistent with the cycle-independent dominance of negative helicity in the same hemisphere (Zhou et al., 2020, and references therein). Event #3 originate from a decayed active region (AR 11882) in the southern hemisphere close to the equator. When it is located near the disk center, the active region exhibits a weak revserse-S geometry manifested by coronal loops in EUV (Figure 9(c)), associated with a filament aligned along the polarity inversion line with a similar geometry, signaling the presence of negative helicity. In contrast, AR 11314, the source region of Event #1 exhibits a forward-S geometry in the east and a reverse-S geomtry in the west (Figure 9(a)), suggesting that the CME probably originates from the west side of the active region. The other two source regions, ARs 11719 (Figure 9(b)) and 12027 (Figure 9(d)), are clearly reverse-S shaped, with the latter being outlined by a sigmoidal filament. Additionally, different aspects of the source region of Event #2, AR 11719, has been studied in detail by several authors (Vemareddy & Mishra, 2015; Vemareddy et al., 2016; Guo et al., 2019).

The ICME’s axial length is unknown but assumed to be lie in the range of 2–$\pi$ AU. At the lower end, the flux rope is so stretched that it is “folded” in half between the Sun and Earth; but at the higher end, the rope is circular (Wang et al., 2015). Alternatively, Hu et al. (2015) compared the lengths of ICME field lines that are derived from energetic electron bursts observed at 1 AU and the magnetic twist derived from the Grad-Shafranov reconstruction method. They concluded that the effective axial lengths of the selected ICME events range between 2 to 4 AU. Unfortunately none of our selected events is associated with in-situ energetic electron bursts. However, the comparison between the number of SASs and the magnetic twist of ICMEs as below may shed new light on their axial lengths.

In the first three events, we found that $\tau$ linearly increases with the increasing number of SASs (left panel of Figure 10). Further, the ratio between the number of SASs over $\tau$ gives a similar length of the flux-rope axis, i.e., $\sim\,$3.5 AU (middle panel of Figure 10), which is slightly above $\pi$ AU, but still falls in the range suggested by Hu et al. (2015). However, the 4th event seem to be an outlier: it is the only event originating from close to the east limb (about $60^{\circ}$ east), but the resultant ICME is still ‘skimmed’ by the near-Earth spacecraft for over 30 hrs (Appendix Figure 14), indicating its large size; the ICME has a moderate $\tau$ but the number of SASs is the largest among the four events, which is translated to an axial length of 8.3 AU. These may be caused by two different but interrelated effects, both resulting from its exceptionally fast speed ($\sim\,$1500 km$\,\text{s}^{-1}$ projected on the plane of sky) as measured in the inner heliosphere by SOHO/LASCO.

First, the ICME experiences a much stronger resistance than the other three events when it ‘plows’ through the solar wind, since in principle the aerodynamic drag force applied on the ICME is proportional to the square of the difference between the ICME speed and the solar wind speed (Cargill, 2004). Hence the drag force acting on the fourth ICME is roughly 3.5 times more than that on the other three, given the solar wind speed in the range of 400–450  $\mathrm{km}\,\mathrm{s}^{-1}$. The resistance restricts the ICME’s radial much more than its lateral expansion (e.g., Manchester et al., 2004). However, this would lead to an unrealistically highly flattened shape (right panel of Figure 10) if the spacecraft crossed the ‘nose’ of the flux rope. We then suggest that the spacecraft actually traverses the leg of the ICME, which is evidenced by its axis being oriented almost along the Sun-Earth line (Table 1). As a comparison, the flux-rope axes in the other three events make large angles with respect to the Sun-Earth line. Hence at the time of the spacecraft passing, the ‘nose’ of the fourth ICME has well passed the Earth, resulting in an axial length much longer than $\pi$ AU. However, if the axis has a circular shape, the nose of the ICME would be over 2.5 AU from the Sun. This is translated to an average propagation speed of $\sim\,$1300 km$\,\text{s}^{-1}$, which is also unrealistic. In reality, we may be witnessing a combination of both the flattening and the leg-passing effect.

Wang et al. (2016) converted the field-line length $L_{\text{mfl}}$ estimated by the electron-probe method in seven magnetic clouds from Kahler et al. (2011) to a twist density $\tau_{e}$ based on the uniformly twisted Gold-Hoyle flux rope model (Gold & Hoyle, 1960), i.e., $L_{\text{mfl}}=L_{a}\sqrt{(1+(2\pi\tau_{e}r)^{2}}$, assuming that the flux-rope axial length $L_{a}$ is in the range of 2–$\pi$ AU. A good correlation is found between $\tau_{e}$ and $\tau$ except for an outlier with an exceptionally large $\tau_{e}$ exceeding 10 turns AU^-1. This could result from an underestimated $L_{a}$, in light of our Event #4, whose estimated axial length greatly exceeds $\pi$ AU.