A Multipeak Solar Flare with a High Turnover Frequency of the Gyrosynchrotron Spectra from the Loop-top Source

The origin of multiple peaks in light curves of various wavelengths remains illusive during flares. Here we discuss the flare of SOL2023-05-09T03:54M6.5 with six flux peaks as recorded by a tandem of new microwave and hard X-ray (HXR) instruments. According to its microwave spectra, the flare represents a high-turnover-frequency (>15 GHz) event. The rather-complete microwave and HXR spectral coverage provides a rare opportunity to uncover the origin of such an event together with simultaneous EUV images. We concluded that (1) the microwave sources originates around the top section of the flaring loops with a trend of source spatial dispersion with frequency; (2) the visible movement of the microwave source from peak to peak originates from the process of new flaring loops appearing sequentially along the magnetic neutral line; (3) the optically thin microwave spectra are hard with the indices (α tn) varying from ∼−1.2 to −0.4, and the turnover frequency always exceeds 15 GHz; (4) higher turnover/peak frequency corresponds to stronger peak intensity and harder optically thin spectra. Using the Fokker–Planck and GX Simulator codes we obtained a good fit to the observed microwave spectra and spatial distribution of the sources at all peaks, if assuming the radiating energetic electrons have the same spatial distribution and single-power-law spectra but with the number density varying in a range of ∼30%. We conclude that the particle acceleration in this flare happens in a compact region nearing the loop-top. These results provide new constraints on the acceleration of energetic electrons and the underlying flare intermittent reconnection process.


INTRODUCTION
Solar flares often manifest multiple peaks in their light curves of microwave and hard X-ray (HXR) emissions.The underlying physics for these peaks remains ambiguous, and may vary from event to event.Such events are often classified as quasi-periodic pulsations (QPP) events, if the peaks present obvious periodicity (e.g.Nakariakov & Melnikov 2009;Zimovets et al. 2021;Li & Chen 2022).
Possible mechanisms include the modulation of emission and energy release by magnetohydrodynamic (MHD) waves (Reznikova et al. 2010;Mclaughlin et al. 2018), the self oscillation caused by steady inflow towards the reconnection site (Nakariakov et al. 2010), and intermittent reconnection (e.g.Li & Gan 2006;Ofman et al. 2011;Liu et al. 2013;Kim et al. 2013;Wu et al. 2016).Ning et al. (2018) investigated the flare of SOL2016-07-24T06:20 with double HXR peaks.They concluded the peaks are due to a two-stage energy release process, with the first peak being non-thermal due to energetic electrons accelerated via the loop-loop reconnection, and the second peak being thermallydominated due to direct heating through the loop-loop reconnection at a relatively high altitude.
This means the peaks may have distinct physical origin.These studies reveal the complex physical nature underlying the radiation peaks of solar flares.
With the CBS and NoRP data, Yan et al. (2023) reported similar spectral properties of flare peaks observed during the X2.2 flare on 2022 April 20.The event has a high turnover frequency (ν t ) of its microwave gyrosynchrotron spectrum that extends from 20 GHz to >40 GHz during the impulsive stage.They reported a power-law dependence of the turnover flux I t on the turnover or peak frequency ν t , and identified the rapid-hardening-then-softening trend within the optically-thin regime of the gyrosynchrotron (GS) radiation.Usual GS spectra of flares present a reversed-V shape, with flux density peaking at ν t that is ∼5-10 GHz for average events.Flares with a high turnover frequency (>15-20 GHz) are of particular interest since this indicates the abundance of mildly-relativistic electrons spiraling within a relatively strong magnetic field (e.g.White et al. 1992;Nagnibeda et al. 2013;Song et al. 2016;Wu et al. 2019).No microwave imaging data are available for the event.This limits further analysis of the radiation sources.
Here we focus on the M6.5 class flare on 2023 May 9 that has been well observed by the instruments mentioned above, with imaging data being available.During its impulsive stage the flare manifests 6 peaks in its microwave and HXR flux curves, all these peaks are characterized by high-turnover frequency.This provides a rare opportunity of further investigation of such event.

Instruments and Data
The microwave images were observed by the low-(3 − 6 GHz) and middle-frequency (6 − 12 GHz) SRH arrays, with the time resolution of ∼3.5 seconds, spatial resolution of 15 − 30 ′′ and 12 − 24 ′′ , and spectral resolution of 0.2 and 0.4 GHz.Note that the spatial resolution only means the smallest details that can be resolved, and the centroid position can be estimated with an error (σ) within a fraction of a beam width (e.g.Condon 1997;Yu et al. 2024) where θ is the half-power beamwidth of the beam size, and SNR is the signal to noise ratio of the synthesized images (typically ∼1000 for SRH).
To minimize the instrumental shifts that may arise in the interferometric images, we further aligned the SRH microwave sources with the features visible in other spectral ranges (in particular, in magnetograms).We created a model with the magnetogram (at 03:48 UT) of the nearby non-flaring active region AR 13297, using the GX Simulator code (Nita et al. 2023) and the technique described by Fleishman et al. (2021).We derived position deviations (i.e., effectively, the differences between the observed and model source centroid positions, because the considered active region was only partially resolved with the SRH) for each frequency with cross-correlation of SRH observations and synthetic images of gyroresonance emission.We then applied the shifts required to remove these deviations (assuming them being constant with time) to all SRH images throughout the flare.The described procedure also allowed us to obtain a more accurate mutual alignment of the SRH images at different frequencies.
The following spectral data were used: 1) the NoRP radio flux densities at 1, 2, 3.75, 9.4, 17, and 35 GHz; 2) the integrated radio flux densities from the flaring region of the above SRH images; 3) the CBS radio flux densities from 35.25 to 39.75 GHz with steps of 500 MHz.

Event Overview
The   In other words, the current event belongs to flares with a high turnover frequency (c.f., Yan et al. 2023).

The Main Peak
We first pay attention to the highest peak (P 2 ) around 03:51:53 UT. Figure 3(a) presents the 94 Å image observed at 03:51:59 UT, overlaid with microwave sources (filled contours).The microwave sources appear between the two UV ribbons (R 1 and R 2 ), being co-spatial with the top-cusp of top section of the flare loop.Such type of the observed microwave brightness distribution, with the maximum brightness close to the looptop can be explained by effects of localized injection and transverse pitch-angle anisotropy of accelerated high-energy electrons as well as by possible high optical thickness of the microwave source in the looptop where magnetic field strength is minimal (e.g.Melnikov et al. 2002;Kuznetsov & Kontar 2015).
We observe some spatial separation among these microwave sources (see filled contours in Fig- ure 3(a)).In the lower-frequency range (From ∼2.8 to ∼6.6 GHz), the sources manifest regular spatial dispersion with frequency, with lower-frequency sources locating to the left and higher-frequency sources locating to the right.The source spacing between 2.8 (orange) and 6.6 GHz (cyan) reaches up to ∼10 ′′ .The higher-frequency sources at 6.6 (cyan), 9.0 (blue), and 10.2 GHz (black) overlap with each other, and are closer to the right ribbon R 2 .These sources together agree with the overall loop top morphology of the flare.The overlap of higher-frequency sources may stem from the projection effect near R 2 .
We further perform a spectral fitting (e.

Other Peaks
In Figure 4(a)-(e), we overplot the microwave contours on EUV images observed around other peaks.
Their characteristics are similar to those of the major peak: 1) the microwave sources are between the corresponding flare loop footpoints; 2) sources at lower frequency present spatial dispersion, while sources at frequency above ∼6.6GHz overlap and are closer to R 2 .
Figure 4(f)-(h) presents the fitted gyrosynchrotron spectra for 9 selected moments (see arrows in Figure 1(c)).During the first peak, the fitted spectra reveal that ν t increases from 12.8 GHz at V 0 to 15.4 GHz at P 1 , and α tn varying from −1.12 at V 0 to −0.75 at P 1 , with increasing flux density.For other peaks, ν t (GHz) and α tn are 15.4 and −0.75 at P 1 , 17.1 and −0.54 at P 3 , 16.0 and −0.57at P 4 , 15.0 and −0.67 at P 5 , and 14.5 and −0.71 at P 6 , respectively.We see that α tn manifests a general decreasing trend from P 3 to P 6 .
We conclude that all these peaks (including the major peak (P 2 )) present similar properties with high turnover frequency (ν t >15 GHz) and hard optically-thin spectra with α tn being larger than -0.8.

Source and spectral evolution
We now analyze the overall evolution of the emission sources during the impulsive stage.Figure 5 shows slope (α tn ) correlate well with the higher corresponding turnover frequency (ν t ).Relation between I t and ν t can be fitted with a power-law dependence of I t ∝ ν 2.05 t , with a correlation coefficient being ∼ 0.72, and that between α tn and ν t can be fitted with α tn ∝ −ν −1.48 t , with a correlation coefficient being ∼ 0.75.
To understand the origin of the observed microwave brightness distribution with the peak closer to the right footpoint, we do modeling of the spatial distribution of nonthermal electrons along a magnetic loop by solving the kinetic Fokker-Planck equation.Here we consider Fokker-Planck equation, which includes nonstationary continuous injection of particles and take into account Coulomb collisions and magnetic mirroring (Hamilton et al. 1990;Reznikova et al. 2009): where f = f (E, µ, s, t) is electron distribution function of kinetic energy E = γ − 1 (in units of mc 2 ), pitch-angle cosine µ = cos α, distance from the flaring loop center s, and time t, S = S(E, µ, s, t) is injection rate, β = v/c, v and c are electron velocity and speed of light, γ = 1/ 1 − β 2 is Lorentz factor, B = B(s) is magnetic field distribution along the loop, λ 0 = 10 24 /n(s)ln Λ, n(s) is plasma density distribution, lnΛ is Coulomb logarithm.
For solving the equation, we follow the initial and boundary conditions suggested in Reznikova et al. (2009).Most of the model parameters are taken to be close to those obtained in Sections 2 and 4.1.
The left part of the model loop with the negative values of s has a stronger magnetic field than the right one, and corresponds to the eastern foot of the observed loop.The injection function S(E, µ, s, t) is supposed to be a product of functions dependent only on one variable (energy E, cosine of pitch angle µ, position s, and time t): S(E, µ, s, t) = S 1 (E) × S 2 (µ) × S 3 (s) × S 4 (t), where the energy dependence is a power law: S 1 (E) = (E/Emin) −δ , E min = 30 keV, with spectral index δ = 2.6 that is equal to one derived from microwave spectrum; S 2 (µ) is a pitch-angle distribution; S 3 (s) is an injection source spatial distribution: S 3 (s) = exp(−(s − s 0 ) 2 /s 2 1 ); S4(t) is a time dependence: S4(t) = exp((t − t m ) 2 /t 2 0 ), t m = 15s, t 0 = 12s, the half-width of injection duration is 30s, similar to a single peak duration of the flare under study.
We have considered three models with the isotropic injection (S 2 (µ) = 1): the first one with the injection location at the looptop (s = 0), the second one with the injection location close to the right footpoint (s = 0.20L), and the third one with the injection close to the left footpoint The first model (Figure 9(a)) presents a single peak of the electron distribution at the location with minimum magnetic field strength (s = 0) in the rising, peak, and decay phases of the flare.This happens because electrons with large pitch-angles are accumulated in the local magnetic trap mostly around B(s) = B min , while electrons with small pitch-angles precipitate into the dense chromosphere.
Two spatially separated peaks appear for the 2 nd and 3 rd models (Figure 9(b) and (c)), with the minimum at the looptop.The reason for this is that energetic electrons with large pitch-angles which originally isotropically injected at the positions in one loop leg have their reflection in the opposite leg where the magnetic field strength is the same (e.g.Kuznetsov et al. 2011;Kuznetsov & Kontar 2015).
The shape of the distributions remains more or less similar in case we increase Coulomb scattering by 10 times, increasing the number density of thermal plasma in the looptop up to N = 5 × 10 10 cm −3 .

Modeling gyrosynchrotron emission distributions
We simulate microwave emission with GX Simulator for the three models with analytical distributions with the shapes similar to the ones obtained from Fokker-Planck simulations: where f 1 , f 2 , and f 3 are the normalized spatial distributions of nonthermal electrons along the flare loop for the 1 st , 2 nd , and 3 rd models (Figure 8(c)).
We show the simulated microwave source distributions in Figure 10.Its comparison with Figure 3(a) shows that the observed source shift to the right footpoint with frequency agrees with the first two models better than the third one.So, we conclude that the electrons are accelerated and injected close to the top of the observed flaring loop, possibly nearing the cusp of reconnecting field lines, in the flare under study.
We further fix the prescribed electron spatial and energy distributions as those used in the 1 st model, but varied the number density n 0 from 2.1×10 8 , 1.9×10 8 , to 1.7×10 8 cm −3 for peaks P 2 , P 3 , and P 4 , respectively.
The modeled sources (upper panels in Figure 11) concentrate around the loop top (an arrow in Figure 11(a)), agreeing with the SRH observations in Figure 3 and 4. The source separations are evident, with low-frequency sources (2.75 (black), 5.01 (blue), and 6.91 (cyan) GHz) nearing the loop top, and high-frequency sources (9.54 (green) and 12.0 (orange) GHz) being closer to the footpoint.
The spacing reaches up to ∼ 5 ′′ between 2.75 and 12.0 GHz.
The modeled spectra also agree with the observations (lower panels in Figure 11).They present typical gyrosynchrotron patterns with large ν t that is ∼19, 18, and 17 GHz for P 2 , P 3 , and P 4 , respectively, being close to those obtained above (see Figures 3 and 4).

SUMMARY
We reported an M6.5 solar flare occurring on 2023 May 9.The flare presents intriguing multi-peak profiles in almost all available flux densities of microwave and HXR.Six local peaks are observed during the impulsive stage, with quite similar characteristics in terms of the microwave spectra and source patterns.
In terms of the microwave images, the SRH sources lie between the two UV footpoint ribbons, corresponding to the looptop/cusp section.For most peaks, the SRH sources at lower frequencies (<∼6.6 GHz) present clear spatial dispersion, while those at higher frequencies overlap with each other.From peak to peak, these sources move from northwest to southeast on the disk over time.
In terms of the microwave spectra, the optically-thin spectral indices (α tn ) vary from ∼ −1.2 to −0.4, and the spectra are very hard in general with a gentle first-hardening-then-softening trend similar to the hard X-ray spectral evolution.Note, however, that there is a difference between δ x and δ of ∼ 2, agreeing with previous studies (reviewed by White et al. 2011).The spectral turnover frequencies (ν t ) of all the temporal peaks remain in the range of ∼ 15 − 22 GHz.The higher turnover frequency correlates with a larger turnover flux density and a harder optically-thin spectrum, with power-law dependence of I t ∝ ν 2.05 t , and α tn ∝ −ν −1.48 t , respectively.These spectral features indicate strong acceleration of high energetic non-thermal electrons around these peaks.
According to our microwave data and GX simulations, the energy spectral index of the nonthermal electrons lies in a range of -1.8 to -2.6.Regarding the spectral turnover frequency, both the Razin effect and the self-absorption effect can affect its value (Razin 1960;Ramaty 1969).In our event, the high turnover frequency ν t and the very-hard optically-thin spectra favours the latter effect since the larger number density of high energy electrons can shift the spectral maximum toward higher frequency, without significantly enhancing their flux density at lower frequencies (see Melnikov et al. 2008;Wu et al. 2019).This is also supported by the found positive correlation between I t and ν t for this event.
The observed microwave features, including spectra, source position and spatial dispersion at different frequencies have been explained using NLFFF extrapolated magnetic flux tube, Fokker-Plank equation solution for electron distribution along the magnetic tube, and GX Simulator simulation for gyrosynchrotron emission.We conclude that the particle acceleration in this flare happens in a compact region in the right portion close to the looptop, possibly in the cusp of reconnecting magnetic field lines.
The features of all observed peaks of emission are similar and differ only in the magnitude of the flux density.It seems that all of them originate from similar flare loops with the same spatial and energy distributions of the nonthermal electrons, but with different electron number density varying in a range of ∼30%.The GS simulations suggest the peaks are generated by nonthermal electrons that concentrate around the right side of the loop-top region.The source separation for each peak at different frequencies is consistent with our GS spatial distribution modeling, according to which stronger magnetic field (being at lower altitude) favours generation of higher-frequency emission (Nindos 2020).
With observations and the simulations, we suggest that the observed multiple peaks during the impulsive stage of this flare stem from intermittent energy release and electron acceleration.The M6.5 flare on 2023 May 9 originated from the NOAA Active Region (AR) 13296 on the solar disk.According to the GOES SXR fluxes (solid lines in Figure 1(a)), the eruption started at ∼03:36 UT, and peaked at ∼03:54 UT.The temporal profiles of both SXR fluxes and their corresponding time derivatives (dashed lines in Figure 1(a)) manifest slight bumps around 03:40 UT, and significant enhancements around 03:54 UT.We can split the event into the pre-impulsive stage (∼03:36-03:46 UT), the impulsive stage (∼03:46-03:55 UT), and the gradual stage (after 03:55 UT).

Figure 2
Figure 2 shows the dynamic evolution with AIA images at 94 Å.During the pre-impulsive stage (Figure 2(a)), two sets of loops (pointed by arrows) meet and reconnect after ∼03:36 UT, generating a large-scale coronal loop system (red arrows in Figure 2(b)).These new loops rise and approach

Figure 1
Figure1presents the microwave and HXR data.According to the microwave temporal profiles, the flux densities above ∼9.4GHz exceed 1000 SFU, and those above 35 GHz reach up to ∼ 3000 SFU (see Figure1(b) and (c)).Six distinct local peaks (black arrows in Figure1(c), marked as P 1 − P 6 ) can be identified from 03:51 to 03:53 UT and appear in all microwave flux curves.The intervals between these peaks are 22s, 17s, 9s, 16s, and 9s, without significant periodicity.This is why we do not classify this flare as a QPP event.The peaks are more prominent at frequencies higher than 10 GHz.The HXR fluxes above ∼100 keV are 1-2 orders of magnitude above the corresponding background values, with similar peaks (Figure1(d)).From Figure1(c) the flux densities at 17 GHz (orange dashed) are close to those at 35 GHz (green dashed) at all local peaks.This indicates that the microwave spectral peak/turnover frequency should lie around or exceed 17 GHz.In other words, the current event belongs to flares with a high turnover g. see function in Ning & Ding 2007; Asai et al. 2013) of the spatially-unresolved nonthermal microwave spectra observed by SRH, NoRP, and CBS.We note that significant error may arise from data discrepancies among instruments due to their different calibration methods.To constrain systematic errors, we cross-calibrate the SRH and CBS data with the NoRP ones measured at the same frequency: we multiply the SRH fluxes with the ratio of NoRP to SRH values at 9.4 GHz, and the CBS fluxes with the ratio of NoRP to CBS values at 35 GHz.The fitted results (see Figure 3(b)) are representative of typical gyrosynchrotron spectra (solid lines) with large turnover frequencies (ν t ) of > 15 GHz (filled circle).At 03:51:34, 03:51:53, and 03:52:01 UT, ν t is ∼15.1, 20.3, and 16.0 GHz, and the turnover (peak) flux I t is ∼1107, 2800, and 2035 SFU, respectively.The optically thin spectra are very hard at the three moments, with index α tn being −0.77, −0.48, and −0.58, respectively, presenting an overall soft-hard-soft spectral pattern.
Figures 6(b) and (c), both the stronger turnover flux density (I t ) and harder optically-thin spectral

(
(s = −0.40L)).The obtained electron number density distributions along the loop for energies of electrons E = 405 keV at different moments of time are shown in Figure 9(a), (b), and (c), respectively.

Figure 2 .
Figure 2. The AIA 94 Å images(a -d), the HMI magnetogram (e), and 1600 Å image (f).The red arrows in panel (a) point at the loops to be reconnected in the pre-impulsive stage, and the red ones in panel (b) point at the just-reconnected loop.The symbol " X " in panel (c) points at the X point of the subsequent reconnection during the impulsive stage.Panel (d) presents the erupting (blue arrows) and post-flare loops.The thick arrow (white) indicates the moving direction of the brightening loop cusps.The two red dashed line in panels (d)-(f) delineate the two UV ribbons (R 1 and R 2 ) and the black dashed line delineates the location of the loop cusps (N L).An animation for AIA 94 Å (for panels (a-d)) from 03:34:47 UT to 04:01:35 UT showing the dynamic evolution of the flare is available.The real-time duration of the animation is 13s.

Figure 4 .
Figure 4. Same as Figure 3 but for the other five peaks (P 1 , P 3 , P 4 , P 5 , and P 6 ).Panels (a)-(e) show EUV images overlaid with the corresponding microwave contours.Panels (f)-(h) show the data and the fitted microwave spectra: (f) for P 1 , (g) for P 3 , and (h) for P 4 , P 5 , and P 6 .

Figure 6 .
Figure 6.Evolution of the microwave spectral parameters from 03:51:00 UT to 03:53:00 UT.Panel (a) presents the fitted optically-thin spectral indices (α tn ), the turnover frequency and the corresponding intensity (ν t and I t ).Panel (b) and (c) presents the fitting of I t and α tn versus ν t , respectively.The dashed line in panels (a) denotes the local peaks.

Figure 7 .
Figure 7.The HXR data of KW (black) and the thick-target spectral fitting with a single power law (red) around the flare peaks (black arrows in Figure 1(c)).The fitted parameters are written.

Figure 8 .
Figure 8. Parameter setup of the GX Simulator.Panel (a) presents the selected flux tube extrapolated with NLFFF extrapolation (line-of-sight view), showing the spatial distribution of nonthermal electrons as the shadowed area.Panel (b) presents the normalized magnetic field strength, and panel (c) presents the normalized density of the nonthermal electrons, along the loop.The arrow in panel (a) and vertical lines in panels (b) and (c) denote the location of the loop top.

Figure 9 .
Figure9.The simulated evolution of nonthermal electron distributions along the flux tube for the 1 st (a), 2 nd (b), and 3 rd (c) models.The lines denote the distributions at 3.0, 7.5, 9.0, 12.0, 18.0, and 30.0s after the electron injection.

Figure 10 .
Figure 10.The simulated microwave source distributions for three models of nonthermal electrons distribution in Figure 8(c): (a)-(c) for the 1 st model, (d)-(f) for the 2 nd model, and (g)-(i) for the 3 rd model.The left, middle and right panels present the modeled images at 2.0, 10.0, and 18.0 GHz, respectively.The maps are not convolved with the instrument beam