## 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 1960s1,2. Subsequently, they are proved to be prevalent at nearly all wavelength bands from radio to γ-rays. Their periods are found to be distributed in a wide range from sub-second to several minutes3.

The mechanisms generating solar flare QPPs can be generally categorized into two groups: magnetohydrodynamic (MHD) waves modulated and oscillatory magnetic reconnection models3,4,5,6,7. 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 mode8,9,10 and kink mode9,11,12, as well as the slow magnetoacoustic mode13,14,15, are common candidates adopted to interpret observed QPPs. The corresponding characteristic period ranges from seconds, minutes to tens of minutes, respectively7.

On the other hand, the oscillatory reconnection model argues that magnetic reconnection plays an essential role in driving most periodic processes of solar flares16,17,18,19. In the standard flare (CSHKP) model20,21,22,23,24,25, 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 emissions26. 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 process7.

The microwave and HXR images provide the key information for diagnosing mechanisms of flare QPPs16,27,28,29,30,31,32,33. 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 emissions27. With the NoRH data, Melnikov et al.28 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.16 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 GHz34, 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 site35,36.

The microwave radioheliograph Expanded Owens Valley Solar Array (EOVSA) in 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 process34,37,38,39. 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 GHz37,38. 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 CS34,40.

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.

## 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 flare34 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 argumentation41,42,43. Considering that the average magnetic field of the CS decreases with height above the primary X point34, 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 strength44,45.

### Quasi-periodicity of microwave sources

With spatially resolved microwave images, we study the temporal variation of the brightness temperature, Tb(t, ν), at each frequency ν. The Tb(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 Tb(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 emissions46,47.

### 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)44,45, 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. 34, 40, 48). 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 easily49, 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 Fig. 2 of ref. 50). 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 simulations51,52,53,54,55. It is expected that waves and shocks generated near the ends of the CS by (even quasi-steady) reconnection outflows56,57 or the quasi-periodic deformation of the CS structure58,59,60 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 s, the SXR emission power at 1–8 Å (PSXR) first grows slowly, then rises rapidly during t = 500–550 s, and reaches its peak near t = 575 s (Fig. 6a). Both the curves of PSXR 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 dPSXR/dt 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.

## 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 observations34,37,38,40,42.

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 wavelengths17,61. 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 emission45. 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 inflows3,4,5,6,7, 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 loops3. Theoretically, the sausage mode can produce expansion/shrinkage of flare loops8, 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 model62,63 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 value8. 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 downflows42,43, 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 from 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 observations64,65,66. 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 sources65. 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 previously48.

## 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 GHz, where n varies from 1 to 30, respectively. The primary beam size of EOVSA is 186. 2/f(GHz) × 35. 2/f(GHz) at the observation time, and we adopt a circular restoring beam 102. 7/f(GHz) to recover microwave images. We perform CLEAN iterations67 for a time-sequence of images at SPWs over 3.4–10.9 GHz, until the peak residual Tb 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 Compo68 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. 69. 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 σ is the expected value of power and ν 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. 68.

### MHD simulation

The numerical simulation in this work is the same as what introduced in ref. 50, in which the effects of anisotropic thermal conduction and chromospheric evaporation are included. The MHD equation we solve is

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

where P* = p + B2/2, $$e=p/\left(\gamma -1\right)+\rho {u}^{2}/2+{B}^{2}/2$$, κ = κ0T2.5, 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 pb = 0.02 and a density distribution

$$\rho \left(y\right)={\rho }_{{{{{{{{\rm{chr}}}}}}}}}+\frac{{\rho }_{{{{{{{{\rm{cor}}}}}}}}}-{\rho }_{{{{{{{{\rm{chr}}}}}}}}}}{2}\left[\tanh \left(\frac{y-{h}_{{{{{{{{\rm{chr}}}}}}}}}}{{l}_{{{{{{{{\rm{tr}}}}}}}}}}+1\right)\right]\,,$$
(3)

where ρcor = 1, ρchr = 105, hchr = 0.1, and $${l}_{{{{{{{{\rm{tr}}}}}}}}}=0.02$$. The simulation starts from a simplified CSHKP CS with a magnetic field profile

$${B}_{x} =0\,,\\ {B}_{y} =\left\{\begin{array}{ll}\sin \left(\frac{\pi x}{2\lambda }\right)\,,& \left|x\right|\,\leqslant\, \lambda \,,\hfill \\ 1\,,& x \, > \, \lambda \,,\hfill \\ -1,& \hfill x \, < \, -\lambda \,,\end{array}\right.\\ {B}_{z} = \sqrt{1-{B}_{x}^{2}-{B}_{y}^{2}}\,,$$
(4)

where λ = 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

$$\eta=\left\{\begin{array}{ll}{\eta }_{{{{{{{{\rm{b}}}}}}}}}+{\eta }_{{{{{{{{\rm{a}}}}}}}}}\exp \left[-\frac{{x}^{2}+{\left(y-{h}_{\eta }\right)}^{2}}{{l}_{\eta }^{2}}\right]\,,& t\,\le\, {t}_{\eta }\,,\\ {\eta }_{{{{{{{{\rm{b}}}}}}}}}\,,& t \, > \, {t}_{\eta }\,,\hfill\end{array}\right.$$
(5)

where, ηa = 5 × 10−4, hη = 0.5, lη = 0.03, tη = 5, and ηb = 5 × 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 L0 = 2.5 × 109 cm, t0 = 51 s, and B0 = 40 G, respectively. The characteristic velocity, namely the background Alfvén speed, is u0 = L0/t0 = 4.90 × 107 cm s−1. Furthermore, we assume that the corona is composed of fully ionized hydrogen which gives the averaged particle mass $$\bar{m}=0.5{m}_{p}=8.36\times 1{0}^{-25}\,{{{{{{{\rm{g}}}}}}}}$$, where mp is the mass of proton. The characteristic mass density can thus be derived as $${\rho }_{0}={B}_{0}^{2}/4\pi {u}_{0}^{2}=5.31\times 1{0}^{-14}\,{{{{{{{\rm{g}}}}}}}}\,{{{{{{{{\rm{cm}}}}}}}}}^{-3}$$, which corresponds to the electron density $${n}_{e0}=0.5{\rho }_{0}/\bar{m}=3.17\times 1{0}^{10}\,{{{{{{{{\rm{cm}}}}}}}}}^{-3}$$. Then, the characteristic pressure, temperature, conductivity, and diffusivity can be respectively deduced as $${p}_{0}={\rho }_{0}{u}_{0}^{2}=127.3\,{{{{{{{\rm{dyn}}}}}}}}\,{{{{{{{{\rm{cm}}}}}}}}}^{-2}$$, $${T}_{0}=\bar{m}{u}_{0}^{2}/{k}_{{{{{{{{\rm{B}}}}}}}}}=14.52\,{{{{{{{\rm{MK}}}}}}}}$$, $${\kappa }_{\parallel 0}={k}_{{{{{{{{\rm{B}}}}}}}}}{\rho }_{0}{u}_{0}{L}_{0}/\bar{m}=1.07 \times 1{0}^{12}\,{{{{{{{\rm{erg}}}}}}}}\,{{{{{{{{\rm{s}}}}}}}}}^{-1}\,{{{{{{{\rm{cm}}}}}}}}^{-1}\,{{{{{{{\rm{K}}}}}}}}^{-1}$$, and η0 = L0u0 = 1.22 × 1017 cm2 s−1, where kB denotes the Boltzmann constant. In this simulation, the ratio of specific heat is γ = 5/3 and the Spitzer conduction coefficient is set as κ0 = 10−6 erg s−1 cm−1 K−3.5. Under this configuration, the total simulation time is 765 s and the duration of fast reconnection, starting at t = 300 s, is comparable with the duration of SXR peak emission, namely $$7\,\min$$ (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 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 $$\left(\nu,\nu+{{{{{{{\rm{d}}}}}}}}\nu \right)$$ is evaluated by ref. 70 as

$${F}_{{{{{{{{\rm{SXR}}}}}}}}}=C{T}^{-1/2}\left(\mathop{\sum}\limits_{i}{n}_{i}{Z}_{i}^{2}\right){n}_{e}\exp \left(-\frac{h\nu }{{k}_{{{{{{{{\rm{B}}}}}}}}}T}\right){g}_{{{{{{{{\rm{ff}}}}}}}}}\left(h\nu,T\right)\,,$$
(6)

where

$$C=\frac{{2}^{5}\pi }{3{m}_{e}{c}^{3}}{\left(\frac{e}{\sqrt{4\pi {\epsilon }_{0}}}\right)}^{6}{\left(\frac{2\pi }{3{k}_{{{{{{{{\rm{B}}}}}}}}}{m}_{e}}\right)}^{1/2}\,,$$
(7)

and

$${g}_{{{{{{{{\rm{ff}}}}}}}}}\left(h\nu,T\right)=\left\{\begin{array}{ll}1,\hfill &h\nu \,\le\, {k}_{{{{{{{{\rm{B}}}}}}}}}T,\\ {\left(\frac{{k}_{{{{{{{{\rm{B}}}}}}}}}T}{h\nu }\right)}^{0.4},\hfill &h\nu \, > \, {k}_{{{{{{{{\rm{B}}}}}}}}}T\,,\end{array}\right.$$
(8)

where e is the elementary charge, me is the mass of electron, c is the speed of light in vacuum, h is the Planck constant, ϵ0 is the permittivity in vacuum, ne is the number density of electrons, and ni and Zi 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

$${P}_{{{{{{{{\rm{SXR}}}}}}}}}={L}_{{{{{{{{\rm{LOS}}}}}}}}}{\int}_{R}{{{{{{{\rm{d}}}}}}}}x{{{{{{{\rm{d}}}}}}}}y\int\nolimits_{{\nu }_{0}}^{{\nu }_{1}}\frac{{F}_{{{{{{{{\rm{SXR}}}}}}}}}}{4\pi }{{{{{{{\rm{d}}}}}}}}\nu \,,$$
(9)

where the integration region, R, is set as $$x\in \left[-12.5,12.5\right]\,{{{{{{{\rm{Mm}}}}}}}}$$ and $$y\in \left[6.25,25\right]\,{{{{{{{\rm{Mm}}}}}}}}$$, and LLOS = 109 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 keV corresponding to the GOES SXR channel of 1–8 Å.