# Microwave Imaging of Quasi-periodic Pulsations at Flare Current Sheet

Yuankun Kou1,2, Xin Cheng1,2,3†, Yulei Wang1,2, Sijie Yu4†, Bin Chen4, Eduard
P. Kontar5 & Mingde Ding1,2
1School of Astronomy and Space Science, Nanjing University, Nanjing 210093,
China
2Key Laboratory of Modern Astronomy and Astrophysics (Nanjing University),
Ministry of Education, Nanjing 210093, China
3Max Planck Institute for Solar System Research, G$\ddot{o}$ttingen, 37077,
Germany
4Center for Solar-Terrestrial Research, New Jersey Institute of Technology,
323 Martin Luther King Jr. Blvd., Newark, NJ 07102-1982, USA
5School of Physics & Astronomy, University of Glasgow, Glasgow, G12 8QQ, UK

Quasi-periodic pulsations (QPPs) are frequently detected in solar and stellar
flares, but the underlying physical mechanisms are still to be ascertained.
Here, we show microwave QPPs during a solar flare originating from quasi-
periodic magnetic reconnection at the flare current sheet. They appear as two
vertically detached but closely related sources with the brighter ones located
at flare loops and the weaker ones along the stretched current sheet. Although
the brightness temperatures of the two microwave sources differ greatly, they
vary in phase with periods of about 10–20 s and 30–60 s. The gyrosynchrotron-
dominated microwave spectra also present a quasi-periodic soft-hard-soft
evolution. These results suggest that relevant high-energy electrons are
accelerated by quasi-periodic reconnection, likely arising from the modulation
of magnetic islands within the current sheet as validated by a 2.5-dimensional
magnetohydrodynamic simulation.

##  I Introduction

Quasi-periodic pulsations (QPPs), the periodic variation of electromagnetic
emissions, have often been observed in celestial environments of different
scales ranging from flaring stars to cosmic-scale gamma ray and fast radio
bursts. In the past decades, flare QPPs from the closest star, the Sun, have
been investigated in detail, with the first one reported in
1960sthompson1962_spectral ; parks1969_sixteen-second . Subsequently, they are
proved to be prevalent at nearly all wavelength bands from radio to
$\gamma$-rays. Their periods are found to be distributed in a wide range from
sub-second to several minutesnakariakov2009_quasi-periodic .

The mechanisms generating solar flare QPPs can be generally categorized into
two groups: magnetohydrodynamic (MHD) waves modulated and oscillatory magnetic
reconnection modelsnakariakov2009_quasi-periodic ; vandoorsselaere2016_quasi-
periodic ; mclaughlin2018_modelling ; kupriyanova2020_quasi-periodic ;
zimovets2021_quasi-periodic . The former explanation attributes flare QPPs to
MHD waves within/among coronal loops, which can periodically modulate
parameters of loops and then result in oscillatory emissions. The induced
sausage modenakariakov2012_sausage ; kolotkov2015_multi-mode ; tian2016_global
and kink modejakimiec2010_investigation ; kolotkov2015_multi-mode ;
nechaeva2019_catalog , as well as the slow magnetoacoustic
modereznikova2011_flare ; kim2012_slow ; kolotkov2018_quasi-periodic , are
common candidates adopted to interpret observed QPPs. The corresponding
characteristic period ranges from seconds, minutes to tens of minutes,
respectivelyzimovets2021_quasi-periodic .

On the other hand, the oscillatory reconnection model argues that magnetic
reconnection plays an essential role in driving most periodic processes of
solar flareshuang2016_quasi-periodic ; kupriyanova2016_relationship ;
hayes2019_persistent ; clarke2021_quasi-periodic . In the standard flare
(CSHKP) modelcarmichael1964_process ; sturrock1966_model ;
hirayama1974_theoretical ; kopp1976_magnetic ; shibata1995_hot-plasma ;
lin2004_role , magnetic reconnection takes place in a vertical current sheet
(CS), which connects flare loops and erupting coronal mass ejections (CMEs),
and accelerates electrons. Once the reconnection process within the CS
presents an oscillatory behavior, which could be either spontaneous or driven
by external MHD waves, electrons will be accelerated quasi-periodically. As a
result, quasi-periodic microwave emissions are generated when accelerated
electrons propagate along magnetic loops. Almost simultaneously, quasi-
periodic hard X-ray (HXR) radiations also present as the electrons reach at
footpoints of the loops, which also explains the similar temporal evolution
pattern between HXR and microwave emissionsdennis1988_solar . Moreover, soft
X-ray (SXR) and extreme-ultraviolet (EUV) emissions in flare loops radiated by
hot plasma from electrons bombarding the chromosphere could also show a quasi-
periodicity. Therefore, in the oscillatory reconnection scenario, quasi-
periodicity of flare emissions at all bands is essentially caused by the
quasi-periodic reconnection processzimovets2021_quasi-periodic .

The microwave and HXR images provide the key information for diagnosing
mechanisms of flare QPPsasai2001_periodic ; melnikov2005_spatially ;
inglis2008_multi-wavelength ; fleishman2008_broadband ; kupriyanova2010_types
; huang2016_quasi-periodic ; kuznetsov2016_spatio-temporal ; li2017_quasi-
periodic . The first imaging observation of flare QPPs with the Hard X-ray
Telescope onboard Yohkoh and the Nobeyama Radioheliograph (NoRH) revealed that
accelerated electrons generate quasi-periodic microwave and HXR
emissionsasai2001_periodic . With the NoRH data, Melnikov et
al.melnikov2005_spatially  found microwave QPPs at different parts of flare
loops and attributed them to the modulation of two oscillation modes in flare
loops. Huang et al.huang2016_quasi-periodic  even clearly resolved the spatial
distribution of the non-thermal microwave QPPs along a limb flare loop.
However, due to the limited frequency coverage (at 17 GHz and 34 GHz), NoRH is
almost impossible to observe the structure of the key energy release region
(reconnection CS) above the flare loops, which is more sensitive to
frequencies lower than 10 GHzchen2020_measurement , and is thus difficult to
distinguish the role of quasi-periodic magnetic reconnection. Besides, the
recent instrument Siberian Radioheliograph and Mingantu Spectral
Radioheliograph also detected microwave QPPs, respectively, but both of which
were found to be not directly related with the main energy release
sitekashapova2021_common ; chen2019_quasi-periodic .

The microwave radioheliograph Expanded Owens Valley Solar Array (EOVSA) came
into use since early 2017, with the state-of-the-art spectro-temporal
resolution capability in microwave, has demonstrated a powerful potentiality
in diagnosing energetic electrons and associated reconnection
processgary2018_microwave ; fleishman2020_decay ; chen2020_measurement ;
fleishman2022_solar . At the time of the observation, it covers a frequency
range of 3–18 GHz with 126 spectral channels combining into 30 spectral
windows (SPWs). At each SPW, high-resolution images can be derived with a high
cadence (up to 1 s). Such a spectral-imaging capability is desired to
construct microwave images of the CS at multiple SPWs and extract broad-
frequency-range spectra at each pixel. It has been documented by recent
observations on the famous 2017 September 10 event, in which the microwave
morphology of the flare loops and the extended CS can be well reconstructed at
frequencies below 10 GHzgary2018_microwave ; fleishman2020_decay . The
spatially resolved microwave spectra obtained at different spatial locations
in the CS are used to derive the magnetic field strength and its distribution
along the CSchen2020_measurement ; chen2021_energetic .

Here, we show microwave imaging of QPPs within the reconnection current sheet
during a C5.9-class solar flare. Simultaneous observations of EOVSA and the
Atmospheric Imaging Assembly (AIA) aboard the Solar Dynamics Observatory (SDO)
reveal that the flare produces two vertically detached but closely related
microwave sources at all wavelength bands from 3.4 to 10.9 GHz, which are
located at the flare loops and extended along a stretched reconnection CS,
respectively. Interestingly, the brightness temperatures and spectral indices
of the microwave spectra in the two sources vary in phase and quasi-
periodically. The 2.5-dimensional MHD simulation suggests the modulation of
magnetic islands within the CS leads to the quasi-periodic magnetic
reconnection and electron acceleration, which then generate microwave QPPs.

##  II Results

Event Overview. The flare on 2017 July 13 is located at active region 12667.
It starts at about 21:46 universal time (UT) and peaks at about 21:55 UT (Fig.
1a). The EOVSA cross-power dynamic spectrum (DS) in the frequency range of
3.4–17.9 GHz clearly shows that some microwave bursts appears repetitively
during the energy release process of the flare. The quasi-periodic emission
extends to the frequency of 18 GHz near the flare peak time (Fig. 1b). The
metric DS observed by the radio spectrometer GREENLAND of the Compound
Astronomical Low frequency Low cost Instrument for Spectroscopy and
Transportable Observatory (e-CALLISTO/GREENLAND) also presents a group of type
III bursts (Fig. 1c). These type III bursts that are believed to be caused by
escaped electrons along the open field seem to also appear quasi-periodically,
well accompanying the microwave bursts, in particular before 21:56 UT. Such a
synchronization implies that the bursts at both wavelengths are most likely of
the same origin, presumably generated by flare-accelerated electrons.

EUV images recorded from the flare impulsive to decay phase show that the
flare is caused by the eruption of a filament (Fig. 1d). At about 21:48 UT,
the filament starts to rise, giving rise to EUV brightenings near its two
footpoints simultaneously. Afterwards, the filament quickly erupts and the SXR
emission rapidly increases. Nevertheless, due to the obscuration of the
filament, the eruption-induced cusp-shaped loops are not observed until after
about 21:57 UT. During the main phase, only two curved flare ribbons are
visible and located at both sides of the remaining filament. After about 23:12
UT, the associated CME appears in the field of view of the Large Angle and
Spectrometric Coronagraph onboard Solar and Heliospheric Observatory.

We measure the height of the erupting filament (red diamonds in Fig. 1e) and
calculate the eruption speed through the first derivative of the height-time
data (Fig. 1f). We find that the speed varies in phase with the Geostationary
Operational Environmental Satellites (GOES) SXR 1–8 Å flux of the flare. With
the SXR flux rising rapidly at about 21:52 UT, the filament speed also quickly
increases, reaching to about 300 km s-1 at the flare peak time. The average
acceleration is about 440 m s-2. During about 21:48 UT–21:52 UT, the filament
also shows a period of acceleration, but the SXR emission only increases
slightly, probably due to the obscuration of the filament.

Microwave and EUV Imaging. Thanks to EOVSA’s spectral-imaging capability at a
high cadence (1 s), the dynamic evolution of the microwave emission sources
during the flare can be determined with a high precision. The 4.4 GHz emission
consists of a primary strong source and a secondary weak source, with the
latter connecting to the main one and extends along the eruption direction
(Fig. 2b). These sources seem to enhance repeatedly but their morphology is
nearly unchanged. To highlight the secondary weak source, we further fit the
main source with a two-dimensional Gaussian model and then subtract the fitted
source from the observed images. The contours of the main and second sources
clearly show that the double-source feature persistently exists at all
observed frequencies (Fig. 2c and Supplementary Movie 1).

A comparison of the microwave sources with the AIA 211 Å images clearly
displays that the strong sources are located at the flare loops and their
centroids at different frequencies are very close to each other. For the weak
sources, however, their centroids ascend along the direction of the eruption
with decreasing frequency (Fig. 2c). These relatively weak microwave sources
closely resemble the observations previously reported in the much stronger
2017 September 10 X8.2 flarechen2020_measurement  and are most likely caused
by non-thermal electrons distributed along a long reconnection CS stretched by
the erupting filament as predicted in the standard model (Fig. 2d). This is
further supported by the presence of reconnection downflows along the CS,
which appear intermittently above and quickly move toward the top of flare
loops after the filament eruption (Supplementary Fig. 1), consistent with
previous argumentationsun2015_extreme ; cheng2018_observations ;
yu2020_magnetic . Considering that the average magnetic field of the CS
decreases with height above the primary X pointchen2020_measurement , the
frequency-dependent distribution of the weak microwave sources observed here
complies with the picture that the peak frequency of the gyrosynchrotron (GS)
emission decreases with a decreasing magnetic field
strengthdulk1982_simplified ; dulk1985_radio .

Quasi-periodicity of microwave sources. With spatially resolved microwave
images, we study the temporal variation of the brightness temperature,
$T_{\mathrm{b}}(t,\nu)$, at each frequency $\nu$. The $T_{\mathrm{b}}(t)$ of
the two sources, both at an optically-thick frequency 3.4 GHz and an
optically-thin frequency 8.4 GHz, increases and decreases repeatedly and in
phase, presenting an evident quasi-periodicity (Fig. 3a,b) and high relation
to each other. Furthermore, the temporal evolution of $T_{\mathrm{b}}(t)$ is
also found to be approximately synchronous with that of the time derivative of
SXR flux, with a time lag of about 13 s (Supplementary Fig. 2). Wavelet
analysis provides the periodicity of these curves. For both the strong and
weak microwave sources at 8.4 GHz, a confident period appears in the range of
about 10–20 s between about 21:55 UT and about 21:56 UT; while the period of
about 30–60 s appears confidently only in the weak sources from about 21:54 to
21:57 UT (Fig. 3c-e). Similarly, the time derivative of SXR flux also presents
a period of about 10–20 s around 21:54 UT as well as between about 21:55 UT
and about 21:56 UT. The synchronization and similar periodicity indicate that
the Neupert effect, referring to the similarity of the temporal evolution
between the microwave/HXR flux and the time-derivative of SXR flux of flares,
applies to this event. It shows that non-thermal electrons producing the HXR
and microwave emissions are the primary heating source of the flare loops
responsible for the SXR emissionsneupert1968_comparison ; veronig2005_physics
.

Spectral properties of microwave sources. The brightness temperature spectra
of the strong and weak but long extended microwave sources, the latter of
which is divided into three small regions along the eruption direction and
labeled as Region 0, 1, and 2 in Fig. 4c, both show a power-law form with a
positive slope at lower frequencies and a negative slope at higher frequencies
(Fig. 4a,b). This is a characteristic of GS emission spectra. Using the power-
law model, we fit the optically-thin part of the spectra (from the peak
frequency to 10.9 GHz, above which the signal-to-noise ratio becomes too low)
and derive the (photon) spectral indices as shown in Fig. 4d. It is found that
these spectral indices are always larger than the typical value for the
optically-thin parts of thermal GS spectra (-10)dulk1982_simplified ;
dulk1985_radio , which indicates that the microwave spectra observed here are
most likely of a non-thermal origin.

We also compare qualitatively the non-thermal properties at different time and
different regions. The spectral indices for microwave sources at the flare
loops and the lower parts of the expected CS (Region 0 and 1) are found to be
largely synchronized with the microwave brightness temperature of the CS
sources at the optically-thin frequency band 8.4 GHz and exhibit a repeated
soft-hard-soft variation behaviour (Fig. 4d and Supplementary Fig. 3).
Comparing with Region 0 and 1, the temporal evolution of the spectral index
for Region 2 seems to be similar but with a delay of about half a minute, in
particular after about 21:54:30 UT (Fig. 4d). The repeated soft-hard-soft
feature detected here implies that electrons responsible for the microwave
sources are accelerated quasi-periodically during the flare. Moreover, the
spectral indices at the different regions are evidently distinct. They are the
hardest (less negative) at Region 0, significantly decrease (softer) toward
Region 1 and 2, but slightly decrease (softer) at the flare loops. This could
be due to that the lowest part of the CS (Region 0) is close to the
reconnection termination shock, where electrons trapped in the magnetic mirror
are accelerated most efficiently. The acceleration is relatively less
efficient in the CS above the termination shock, even further weakening with
the increase of the height such as from Region 0 to 2 (also see
Ref.chen2015_particle ; chen2020_measurement ; chen2021_energetic ). While for
the microwave spectra at the flare loops, they are likely generated by escaped
electrons from the acceleration region above the flare loop-top. Together with
the fact that electrons with lower energies escape more
easilymelrose1976_precipitation , the spectral indices are thus smaller
(softer) than that at Region 0.

MHD simulation of quasi-periodic magnetic reconnection. To explore the origin
of quasi-periodic energy release during the flare reconnection, we run a high-
resolution 2.5-dimensional resistive MHD simulation. We find that, during the
fast reconnection, magnetic islands are quasi-periodically generated near
X-points in the main CS. The islands move downwards and then interact with the
flare loops quasi-periodically (Fig. 5a-c). Owing to the modulation of the
islands, the energy release (or reconnection) rate presents a quasi-
periodicity (see Figure 2 of Ref.wang2021_annihilation ). Such a quasi-
periodic energy release process may accelerate electrons quasi-periodically,
thus generating the QPPs at the microwave bands that we have observed. In
fact, the quasi-periodic characteristics related to the generation of islands
in the CS has been revealed by numerous previous MHD
simulationskliem2000_solar ; bhattacharjee2009_fast ; huang2010_scaling ;
guidoni2016_magnetic-island ; zhao2020_mesoscale . It is expected that waves
and shocks generated near the ends of the CS by (even quasi-steady)
reconnection outflowstakasao2016_above-the-loop-top ; takahashi2017_quasi-
periodic  or the quasi-periodic deformation of the CS
structuremurray2009_simulations ; mclaughlin2012_periodicity ;
thurgood2017_three-dimensional  could also lead to oscillations of various
flare structures and emissions.

Based on the simulated density and temperature distributions, we also
calculate the synthetic thermal X-ray emission. We find that, starting from
$t=300\,\mathrm{s}$, the SXR emission power at 1–8 Å($P_{\mathrm{SXR}}$) first
grows slowly, then rises rapidly during $t=500$–$550\,\mathrm{s}$, and reaches
its peak near $t=575\,\mathrm{s}$ (Fig. 6a). Both the curves of
$P_{\mathrm{SXR}}$ and its time derivative show quasi-periodic features
similar to the GOES SXR lightcurve and its time derivative, respectively (Fig.
3 and Fig. 6a-c). Similar to the observed periodicity, the wavelet analysis of
$\mathrm{d}P_{\mathrm{SXR}}/\mathrm{d}t$ also presents two primary periodic
components, with the stronger one of about 16–30 s and the weaker other of
about 30–60 s (Fig. 6c). Especially, the number distribution of the magnetic
islands as a function of magnetic flux and time in the integration region
shows that the formation of the QPPs is closely related to the generation of
magnetic islands (Fig. 6d). In short, the comparability between sythetic and
observed periodicity supports that the oscillatory magnetic reconnection is
the main cause of the QPPs as detected for the current event.

##  III Discussion

In this work, we analyze an eruptive flare on 2017 July 13 focusing on its
spatially resolved microwave emissions observed by EOVSA. Combining EUV images
provided by SDO/AIA, we find that the microwave emissions are mainly from two
vertically detached but closely related sources. The main strong sources are
located at the flare loops with their centroids almost unchanged at different
frequencies. For the second weak ones, they extend toward the direction of the
filament eruption with their centroids ascending with decreasing frequency.
This provides strong evidence that the microwave sources are emitted by non-
thermal electrons accelerated in the close vicinity of the flare CS as found
by recent observationscheng2018_observations ; gary2018_microwave ;
fleishman2020_decay ; chen2020_measurement ; chen2021_energetic .

The most striking finding is that the brightness temperature and spectra for
the two CS-associated microwave sources vary synchronously and present an
obvious quasi-periodicity. Through wavelet analysis, we find that the main
period is in the range of about 10–20 s and the second period in the range of
about 30–60 s. Such periodicity is also observed in the time derivative of the
spatially integrated SXR emission. The periods observed here are basically
compatible with previous results obtained in other events and at different
wavelengthskupriyanova2016_relationship ; zhang2016_chromospheric . Moreover,
the similarity between the evolution of the microwave brightness temperature
and the time derivative of SXR flux hints the major role of non-thermal
electrons in heating the flare plasma and exciting the corresponding
emissions. We also find that the optically-thin parts of the spectra tend to
be flatter (harder) than those for the typical thermal GS
emissiondulk1985_radio . We suggest that the quasi-periodic flare reconnection
during the event drives the acceleration of the non-thermal electrons and then
produces quasi-periodic microwave emissions at both the reconnection CS and
the flare loops. Our 2.5-dimensional MHD numerical simulation interprets that
the quasi-periodicity of the magnetic reconnection is caused by the appearance
of magnetic islands within the long-stretched CS, which is further testified
by the similarity between the periods of the synthetic and observed SXR
emission. It is also possible that the quasi-periodic reconnection is caused
by an external driver that quasi-periodically changes the reconnection
inflowsnakariakov2009_quasi-periodic ; vandoorsselaere2016_quasi-periodic ;
mclaughlin2018_modelling ; kupriyanova2020_quasi-periodic ;
zimovets2021_quasi-periodic , which, however, is hardly detected in the
current event.

The periodicity of the QPPs we derived seems to support the interpretation of
the sausage mode, which is also able to generate flare QPPs at the microwave
bands via modulating the magnetic field strength and the electron density
within flare loopsnakariakov2009_quasi-periodic . Theoretically, the sausage
mode can produce expansion/shrinkage of flare loopsnakariakov2012_sausage ,
but which is not observable for this event even with EUV images of the high
spatio-temporal resolution. Moreover, the QPP phase difference between high
(optically-thin bands) and low frequencies (optically-thick bands) predicted
in the MHD modelmossessian2012_modeling ; kuznetsov2015_simulations  is also
absent (see Fig. 3a,b). Thus, we tend to exclude the possibility of sausage
mode giving rise to the QPPs as detected along the CS. Of course, the sausage
mode in the flare loops could leak out when its wavelength exceeds a cutoff
valuenakariakov2012_sausage . But this is essentially equivalent to external
MHD waves giving rise to the quasi-periodic reconnection within the CS.

The spatially resolved EOVSA data clearly locate the sources of the flare
QPPs, in particular the CS sources above the flare loops, which provides
convincing evidence for quasi-periodic magnetic reconnection as a cause of
quasi-periodic flare emissions. Due to the quasi-periodicity of the
reconnection, electrons are accelerated quasi-periodically either in the
reconnection CS or in the termination shock above the flare loop-top, later of
which is driven by quasi-periodic plasma downflows(cheng2018_observations, ;
yu2020_magnetic, ), and produce quasi-periodic microwave emissions at the
reconnection region. As the energetic electrons are injected quasi-
periodically into the flare loops, they also excite quasi-periodic microwave
emissions there. Moreover, the flare also produces a group of quasi-periodic
type III bursts, the peaks of which coincide well with that of the microwave
emissions. This supports the speculation that energetic electrons originating
from the quasi-periodic reconnection also produce quasi-periodic type III
bursts once they escape the reconnection region and propagate away from the
Sun.

The separation of the two microwave sources in the eruption direction is
similar to but largely different from that of double HXR sources in the corona
discovered by Ramaty High Energy Solar Spectroscopic Imager
observationssui2004_evidence ; liu2008_double ; chen2017_double-coronal . In
the framework of the standard flare model, the two vertically separated HXR
sources are interpreted as those formed by the interactions of two oppositely
directed reconnection outflows with flare loops and the erupting flux rope
structure, respectively. The upper coronal HXR sources that rise with the
eruption usually have softer spectra than the lower sourcesliu2008_double . On
the contrary, the microwave sources at the CS in our event do not
significantly change their locations with time, and their lowest part has a
harder spectral index than that of the flare-loop sources. These facts further
support that the upper microwave sources are along the stretched reconnection
CS with the lower tip probably corresponding to the termination shock as
argued previouslychen2015_particle .

##  IV Methods

Microwave data reduction and spectral analysis. At the time of the
observation, EOVSA has 126 spectral channels covering a frequency range of
about 3–18 GHz, which combine into 30 SPWs to increase the dynamic range when
imaging (so-called multifrequency synthesis technique). The band width of each
SPW is 500 MHz, and central frequencies are $f=2.92+n/2\,\mathrm{GHz}$, where
$n$ varies from 1 to 30, respectively. The primary beam size of EOVSA is
$186^{\prime\prime}.2/f\mathrm{(GHz)}\times
35^{\prime\prime}.2/f\mathrm{(GHz)}$ at the observation time, and we adopt a
circular restoring beam $102^{\prime\prime}.7/f\mathrm{(GHz)}$ to recover
microwave images. We perform CLEAN iterationshogbom1974_aperture  for a time-
sequence of images at SPWs over 3.4–10.9 GHz, until the peak residual
$T_{\mathrm{b}}$ down to a level of about $10^{-1}$ MK and thus the signal-to-
noise ratio, in most cases, higher than 10. Tools for all these processes are
included in the CASA (https://casa.nrao.edu) and SunCASA
(https://github.com/suncasa/suncasa-src) packages.

High-resolution microwave images at multiple SPWs make it possible to obtain
spatially-resolved microwave spectra at each time. We fit the optically-thin
part of the spectra with the power-law model to get the spectral indices at
different instants or regions. In practice, the optically-thin part is defined
as the frequency range from the peak frequency, of each specific spectrum, to
10.9 GHz. For some spectra, if less than four frequency points remain within
the optically-thin part (typically because of the relatively low spectra
quality), they are not fitted. To reduce such spectra, an 11-second smoothing
is applied.

Wavelet analysis. We utilize the wavelet analysis to quantify the periodicity
of the flare microwave emissions. The core wavelet software is provided by
Torrence and Compotorrence1998_practical  and available at
http://atoc.colorado.edu/research/wavelets ; the code for fitting the
background power spectra and determining the significance levels is adopted
from Ref.auchere2016_fourier . Here, we use the mother wavelet ‘Morlet’ and
fit the background Hann-window-apodized Fourier spectra with the combination
of a power-law function and a constant value as

$\sigma(\nu)=A\nu^{s}+C\,,$ (1)

where $\sigma$ is the expected value of power and $\nu$ is frequency, to
determine the 95% significance levels. We choose the 95% global significance
level to confirm the periodicity, which takes into account the total number of
degrees of freedom in the wavelet power spectra and is argued to be more
reliable than the 95% local significance level traditionally used by
Ref.torrence1998_practical .

MHD simulation. The numerical simulation in this work is the same as what
introduced in Ref.wang2021_annihilation , in which the effects of anisotropic
thermal conduction and chromospheric evaporation are included. The MHD
equation we solve is

$\displaystyle\frac{\partial\rho}{\partial t}+\nabla\cdot\left(\rho{\bf
u}\right)$ $\displaystyle=$ $\displaystyle 0\,,$
$\displaystyle\frac{\partial\left(\rho{\bf u}\right)}{\partial
t}+\nabla\cdot\left(\rho{\bf u}{\bf u}-{\bf BB}+P^{*}{\bf I}\right)$
$\displaystyle=$ $\displaystyle 0\,,$ $\displaystyle\frac{\partial e}{\partial
t}+\nabla\cdot\left[\left(e+P^{*}\right){\bf u}-{\bf B}\left(\bf{B}\cdot{\bf
u}\right)\right]$ $\displaystyle=$
$\displaystyle\nabla\cdot\left(\kappa_{\parallel}\hat{\bf{b}}\hat{\bf{b}}\cdot\nabla
T\right)\,,$ (2) $\displaystyle\frac{\partial{\bf B}}{\partial
t}-\nabla\times\left(\bf{u}\times\bf{B}\right)$ $\displaystyle=$
$\displaystyle-\nabla\times\left(\eta{\bf J}\right)\,,$ $\displaystyle{\bf J}$
$\displaystyle=$ $\displaystyle\nabla\times{\bf B}\,,$

where $P^{*}=p+B^{2}/2$, $e=p/\left(\gamma-1\right)+\rho u^{2}/2+B^{2}/2$,
$\kappa_{\parallel}=\kappa_{0}T^{2.5}$, ${\bf I}$ is the identity matrix, and
standard notations of variables are used. All quantities are dimensionless in
our simulation. We suppose an initial atmosphere with a uniform background
pressure $p_{\mathrm{b}}=0.02$ and a density distribution

$\rho\left(y\right)=\rho_{\mathrm{chr}}+\frac{\rho_{\mathrm{cor}}-\rho_{\mathrm{chr}}}{2}\left[\mathrm{tanh}\left(\frac{y-h_{\mathrm{chr}}}{l_{\mathrm{tr}}}+1\right)\right]\,,$
(3)

where $\rho_{\mathrm{cor}}=1$, $\rho_{\mathrm{chr}}=10^{5}$,
$h_{\mathrm{chr}}=0.1$, and $l_{\mathrm{tr}}=0.02$. The simulation starts from
a simplified CSHKP CS with a magnetic field profile

$\displaystyle B_{x}$ $\displaystyle=$ $\displaystyle 0\,,$ $\displaystyle
B_{y}$ $\displaystyle=$ $\displaystyle\begin{cases}\sin\left(\frac{\pi
x}{2\lambda}\right)\,,&\left|x\right|\leqslant\lambda\,,\\\
1\,,&x>\lambda\,,\\\ -1\,,&x<-\lambda\,,\end{cases}$ (4) $\displaystyle B_{z}$
$\displaystyle=$ $\displaystyle\sqrt{1-B_{x}^{2}-B_{y}^{2}}\,,$

where $\lambda=0.1$ is the half-width of the CS. A localized anomalous
resistivity is initially applied to trigger the fast reconnection, which is
expressed by

$\eta=\begin{cases}\eta_{\mathrm{b}}+\eta_{\mathrm{a}}\mathrm{exp}\left[-\frac{x^{2}+\left(y-h_{\eta}\right)^{2}}{l_{\eta}^{2}}\right]\,,&t\leq
t_{\eta}\,,\\\ \eta_{\mathrm{b}}\,,&t>t_{\eta}\,,\end{cases}$ (5)

where, $\eta_{\mathrm{a}}=5\times 10^{-4}$, $h_{\eta}=0.5$, $l_{\eta}=0.03$,
$t_{\eta}=5$, and $\eta_{\mathrm{b}}=5\times 10^{-6}$. The boundary condition
is set as follows. The left ($x=-1$) and right ($x=1$) are free boundaries,
the top ($y=4$) is a no-inflow boundary, and the bottom ($y=0$) is a symmetric
boundary. The above system is simulated with Athena 4.2. The conservation part
of MHD equation is solved by the HLLD Riemann solver, the 3-order piece-wise
parabolic flux reconstruction algorithm, and the Corner Transport Upwind (CTU)
method. The resistivity and TC are calculated by the explicit operator
splitting method. We set a high-precision uniform Cartesian mesh with 3840 and
7680 grids on $x$ and $y$ directions, respectively.

Here, we use new characteristic parameters for normalization based on the
space and time scales of the observed flare in order to compare with results
derived from observations. The characteristic length, time, and magnetic field
are set as $L_{0}=2.5\times 10^{9}\,\mathrm{cm}$, $t_{0}=51\,\mathrm{s}$, and
$B_{0}=40\,\mathrm{G}$, respectively. The characteristic velocity, namely the
background Alfvén speed, is $u_{0}=L_{0}/t_{0}=4.90\times
10^{7}\,\mathrm{cm\,s^{-1}}$. Furthermore, we assume that the corona is
composed of fully ionized hydrogen which gives the averaged particle mass
$\bar{m}=0.5m_{p}=8.36\times 10^{-25}\,\mathrm{g}$, where $m_{p}$ is the mass
of proton. The characteristic mass density can thus be derived as
$\rho_{0}=B_{0}^{2}/4\pi u_{0}^{2}=5.31\times 10^{-14}\,\mathrm{g\,cm^{-3}}$,
which corresponds to the electron density
$n_{e0}=0.5\rho_{0}/\bar{m}=3.17\times 10^{10}\,\mathrm{cm^{-3}}$. Then, the
characteristic pressure, temperature, conductivity, and diffusivity can be
respectively deduced as
$p_{0}=\rho_{0}u_{0}^{2}=127.3\,\mathrm{dyn\,cm^{-2}}$,
$T_{0}=\bar{m}u_{0}^{2}/k_{\mathrm{B}}=14.52\,\mathrm{MK}$, $\kappa_{\parallel
0}=k_{\mathrm{B}}\rho_{0}u_{0}L_{0}/\bar{m}=1.07\times
10^{12}\,\mathrm{erg\,s^{-1}\,cm^{-1}\,K^{-1}}$, and
$\eta_{0}=L_{0}u_{0}=1.22\times 10^{17}\,\mathrm{cm^{2}\,s^{-1}}$, where
$k_{\mathrm{B}}$ denotes the Boltzmann constant. In this simulation, the ratio
of specific heat is $\gamma=5/3$ and the Spitzer conduction coefficient is set
as $\kappa_{0}=10^{-6}\,\mathrm{erg\,s^{-1}\,cm^{-1}\,K^{-3.5}}$. Under this
configuration, the total simulation time is $765\,\mathrm{s}$ and the duration
of fast reconnection, starting at $t=300\,\mathrm{s}$, is comparable with the
duration of SXR peak emission, namely $7\,\mathrm{min}$ (21:53–22:00 UT), of
the flare. Meanwhile, at the end of the simulation, the footpoint separation
of the flare loops is about $10\,\mathrm{Mm}$, which is also comparable with
that of the real flare observed at AIA 131 Å after 22:00 UT (Fig. 1d and Fig.
5d).

The synthetic SXR power density in the frequency window
$\left(\nu,\nu+\text{d}\nu\right)$ is evaluated by Ref.zhao2019_forward  as

$F_{\mathrm{SXR}}=CT^{-1/2}\left(\sum_{i}n_{i}Z_{i}^{2}\right)n_{e}\exp\left(-\frac{h\nu}{k_{\mathrm{B}}T}\right)g_{\mathrm{ff}}\left(h\nu,T\right)\,,$
(6)

where

$C=\frac{2^{5}\pi}{3m_{e}c^{3}}\left(\frac{e}{\sqrt{4\pi\epsilon_{0}}}\right)^{6}\left(\frac{2\pi}{3k_{\mathrm{B}}m_{e}}\right)^{1/2}\,,$
(7)

and

$g_{\mathrm{ff}}\left(h\nu,T\right)=\begin{cases}1,&h\nu\leq
k_{\mathrm{B}}T,\\\
\left(\frac{k_{\mathrm{B}}T}{h\nu}\right)^{0.4},&h\nu>k_{\mathrm{B}}T\,,\\\
\end{cases}$ (8)

where $e$ is the elementary charge, $m_{e}$ is the mass of electron, $c$ is
the speed of light in vacuum, $h$ is the Planck constant, $\epsilon_{0}$ is
the permittivity in vacuum, $n_{e}$ is the number density of electrons, and
$n_{i}$ and $Z_{i}$ are the number density and charge number of ions of
element $i$, respectively. Here, for simplicity, we assume that the plasma is
composed of pure hydrogen. The X-ray emission from a two-dimensional region,
enclosing the reconnection sites in the main CS and the flare loops, at the
frequency window $\left[\nu_{0},\nu_{1}\right]$ is thus calculated by

$P_{\mathrm{SXR}}=L_{\mathrm{LOS}}\int_{R}\mathrm{d}x\mathrm{d}y\int_{\nu_{0}}^{\nu_{1}}\frac{F_{\mathrm{SXR}}}{4\pi}\mathrm{d}\nu\,,$
(9)

where the integration region, $R$, is set as
$x\in\left[-12.5,12.5\right]\,\mathrm{Mm}$ and
$y\in\left[6.25,25\right]\,\mathrm{Mm}$, and
$L_{\mathrm{LOS}}=10^{9}\,\mathrm{cm}$ is the estimated scale of the CS along
the line of sight (LOS). The frequency window is determined by the energy
range of $1.6$–$12.4\,\mathrm{keV}$ corresponding to the GOES SXR channel of
1–8 Å.

Data Availability

The datasets generated during and/or analysed during the current study are
available from the corresponding author upon request. The SDO/AIA data are
also available at http://jsoc.stanford.edu/ajax/lookdata.html, EOVSA data from
http://ovsa.njit.edu/data-browsing.html, GOES data from
https://www.swpc.noaa.gov/products/goes-x-ray-flux (GOES), and e-CALLISTO data
from http://soleil.i4ds.ch/solarradio/callistoQuicklooks/?date=20170713.

Code Availability

EOVSA data processing tools are included in the open-source SunCASA package
(https://github.com/suncasa/suncasa-src), which is based on the CASA
(https://casa.nrao.edu). Codes for processing GOES and SDO/AIA data are
included in the SolarSoftWare (SSW) repository
(https://www.lmsal.com/solarsoft/). E-CALLISTO processing codes are available
at https://www.e-callisto.org. The core wavelet analysis software is provided
by Torrence and Compotorrence1998_practical  and available at
http://atoc.colorado.edu/research/wavelets ; the open-source code for fitting
the background power spectra and determining the significance levels is
available at https://idoc.ias.u-psud.fr/MEDOC/wavelets_tc98. The MHD code is
accessible at https://princetonuniversity.github.io/Athena-Cversion/.

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Acknowledgements Y.K., X.C., Y.W. and M.D. are funded by NSFC grants 11722325,
11733003, 11790303 and by National Key R&D Program of China under grant
2021YFA1600504. X.C. is also supported by Alexander von Humboldt foundation.
B.C. and S.Y. acknowledge support by US NSF grants AGS-1654382 and AST-2108853
to NJIT. EOVSA operations are supported by US NSF grants AST-1910354 and
AGS-2130832 to NJIT.

Author Contributions X.C. led the project. Y.K. and S.Y. analysed the data.
Y.W. performed MHD simulations. X.C. and Y.K. wrote the manuscript. B.C.,
E.P.K and M.D. took participate in the discussion and revised the manuscript.

Competing Interests The authors declare no competing interests.

Correspondence and requests for materials should be addressed to Xin Cheng
<EMAIL_ADDRESS>and Sijie Yu<EMAIL_ADDRESS>

Fig. 1: Overview of the flare event. a. GOES SXR 1–8 Å flux curve of the
flare. b and c. EOVSA cross-power and e-CALLISTO/GREENLAND DS during
21:53-22:00 UT. d. SDO/AIA composite images of 131 (blue), 171 (red) and 211 Å
(green) at 21:54:58 UT and 22:09:22 UT. The white dotted lines represent slits
along the direction of the filament eruption. e. AIA 211 Å slice-time plot
showing the evolution of the filament top. The diamond symbols in red
represent the measured heights, and the white dash lines indicate the five
instants of snapshots in Fig. 2. f. Temporal evolution of the filament speed
(black) and the GOES SXR 1–8 Å flux (blue) of the associated flare. The speed
errors are from the uncertainty in height measurements (5 pixels, about 2.14
Mm).

Fig. 2: EUV and microwave emissions of the flare. a. SDO/AIA 211 Å images
showing the erupting filament and induced EUV brightenings. b. EOVSA microwave
images displaying the distribution and evolution of the flare brightness
temperature, $T_{\mathrm{b}}$, at 4.4 GHz. c. SDO/AIA 211 Å images with
superimposed microwave contours. The color from purple to yellow correspond to
frequencies from 3.4 GHz to 10.4 GHz. There are two groups of contours, each
of which shows 80% of the highest $T_{\mathrm{b}}$ in the strong and weak
sources, respectively. d. A schematic diagram interpreting the magnetic field
structure during the eruption. The wide curve in grey represents the erupting
filament with the overlying field in gold and purple. The ovals in shallower
and deeper grey represent magnetic positive and negative polarities,
respectively. The circles in blue and red, from shallower to deeper with
decreasing frequency, indicate the strong and weak microwave sources locating
at the flare loops and reconnection CS region, respectively.

Fig. 3: Temporal variations and periodicity of brightness temperature and time
derivative of SXR flux. a. The variations of brightness temperature,
$T_{\mathrm{b}}$, the maximal values within corresponding boxes as in Fig. 4c,
of the microwave sources at the flare loops (blue, Region C) and the entire CS
region (red, including Region 0, 1, and 2) at the frequency of 3.4 GHz. The
black curves represent the 11-second-smoothed time derivative of the GOES SXR
1–8 Å emission. The vertical dot lines mark the five instants when the
resolved microwave and EUV images are shown in Fig. 2. b. The same as a but at
8.4 GHz. c. The normalized wavelet power spectrum of $T_{\mathrm{b}}$
variation of the flare loops region at 8.4 GHz and time-averaged
Fourier/wavelet spectrum. The region with shallower color denotes the cone of
influence (COI), under which the periods are insignificant. The black dotted
contours enclose the regions with local wavelet significance higher than 95%,
and the solid contours enclose the regions with global wavelet significance
higher than 95%. Fitting results of the Fourier spectrum (histogram) are
plotted with the blue solid line, with mean power components in blue dotted
lines. The thick black line is the average wavelet spectrum. The gray solid
line marks the 95% significance level of the Fourier spectrum, the red dotted
line the 95% local time-averaged significance level, and the red solid line
the 95% global time-averaged significance level. Fitted power-law index ($s$,
see Formula 1) and the least-square value ($\chi^{2}$) are shown in the lower
right corner of the panel. d and e. The same as c but for the wavelet analysis
of the $T_{\mathrm{b}}$ variation in the entire CS region at 8.4 GHz, and of
the time derivative of SXR flux of the flare, respectively.

Fig. 4: Spatially resolved microwave spectra and their temporal evolution.
Microwave spectra for the flare loops (Region C; a) and the lower part of the
CS (Region 0; b) in the frequency range of 3.4–10.9 GHz. The brightness
temperatures, $T_{\mathrm{b}}$, refer to maximal values in the corresponding
boxes as shown in panel c. The errors are the root-mean-squares of residual
images, which are derived by subtracting two-dimensional Gaussian fitted
microwave sources from the observed ones. c. The AIA 211 Å image overlaid by
the contours of microwave sources at 21:55:37 UT. Boxes with different colors
show the regions for deriving the microwave spectra. d. Temporal evolution of
spectral indices (11-second-smoothed) in the frequency range from the peak to
10.9 GHz for the flare loops (Region C) and the stretched reconnection region
(Region 0, 1 and 2), with the errors indicated by translucent regions. The
errors are the standard deviations. The black curve represents the variation
of the maximal $T_{\mathrm{b}}$ in the entire CS region at the frequency of
8.4 GHz, and the vertical dot lines the five instants shown in Fig. 2a.

Fig. 5: Temperature distributions of the CS at four typical instants. a, b and
c. The CS is fragmented into a number of magnetic islands. A relatively large
magnetic island is interacting with the flare loop-top and several islands in
the main CS are moving downwards. The gray curves mark the magnetic field
lines. d. The spatial scale of the flare loops becomes comparable with
observations at later time.

Fig. 6: Quasi-periodic characteristics of synthetic SXR variation. a. Temporal
evolution of the SXR emission power at 1–8 Å, $P_{\mathrm{SXR}}$. b. Temporal
evolution of the time derivative of $P_{\mathrm{SXR}}$,
$\mathrm{d}P_{\mathrm{SXR}}/\mathrm{d}t$. c. The wavelet power spectrum of the
temporal evolution of $\mathrm{d}P_{\mathrm{SXR}}/\mathrm{d}t$. Same as in
Fig. 3, the gray shading denotes the cone of influence (COI), and the black
contours enclose the regions where the global (solid line) and local (dashed
line) significance is higher than 95%. d. The evolution of the number
distribution function of the magnetic island flux, $N\left(\psi,t\right)$, in
the integration region, where $\psi$ is the flux of magnetic islands and
$\psi_{0}=10^{11}\,\mathrm{Mx\ cm^{-1}}$. $N\left(\psi,t\right)$ is calculated
by the same method as in Ref.wang2021_annihilation .