# Common origin of quasi-periodic pulsations in microwave and decimetric solar
radio bursts

Larisa K. Kashapova  Dmitrii Y. Kolotkov  Elena G. Kupriyanova  Anastasiia V.
Kudriavtseva  Chengming Tan  Hamish A. S. Reid  Institute of Solar-Terrestrial
Physics, SB Russian Academy of Sciences, 664033 Irkutsk, Russia  Centre for
Fusion, Space and Astrophysics, Physics Department, University of Warwick,
Coventry CV4 7AL, United Kingdom  Central Astronomical Observatory at Pulkovo,
Russian Academy of Sciences, 196140 Saint Petersburg, Russia  CAS Key
Laboratory of Solar Activity, National Astronomical Observatories of Chinese
Academy of Sciences, Datun Road A20, Chaoyang District, Beijing 100101, China
Department of Space and Climate Physics, University College, London, RH5 6NT,
UK  School of Physics and Astronomy, University of Glasgow, Glasgow G12 8QQ,
UK

###### keywords:

Flares; Flares, Waves; Radio Bursts, Association with Flares; Radio Emission,
Active Regions; Waves, Magnetohydrodynamic

##  1 Introduction

s:Introduction

Flares are the most complicated and powerful phenomena in the solar
atmosphere. Flare energy release occurs in the corona but the effects can
extend down to the photosphere, and out to the upper corona and interplanetary
space. Such a wide range of heights with significantly different plasma
conditions results in electromagnetic emission ranging from visible and UV to
radio, X- and $\gamma$-ray spectral bands (see, for example, Fletcher et al.,
2011). Despite different energetics and generation mechanisms, manifestations
of solar flares in different spectral bands are undoubtedly related. However,
the specific physical mechanisms of these relationships are still unclear and
remain an open and actively debated research topic. One of the most widely
known examples of such correlations between the multi-band flare emissions is
the empirically established Neupert effect: The connection between the non-
thermal X-ray and microwave emission time profiles, and the derivative of the
thermal soft X-ray time profiles (Neupert, 1968). Another correlation example
is the solar radio type III bursts in the meter radio band and decimetric
spike bursts. Both radio bursts are associated with electrons accelerated
during flares or flare-like events. There are two possible origins of the
accelerated electrons that produce this radio emission. Electrons can be
accelerated by the primary flare energy release and travel along open magnetic
field lines where they can escape the solar atmosphere. Alternatively,
accelerated electrons can be produced in smaller-scale, local current sheets
formed by interacting magnetic structures in the same active region. For
example, Battaglia and Benz (2009) and Benz, Battaglia, and Vilmer (2011)
demonstrated that the position of decimetric emission sources did not agree
with the model of electrons trapped in a flare loop. The generation of
accelerated electrons by current sheets was suggested as a more realistic
scenario. On the other hand, it raises the question of the connection between
the decimetric emission produced by these electrons and the flare emission in
the microwave and X-ray bands.

The microwave (MW) radiation from solar flares is usually associated with
gyrosynchrotron emission from relativistic (or nearly relativistic) electrons.
It is an incoherent emission at frequencies usually above 1 GHz, controlled by
the accelerated electron properties and the strength of the magnetic field in
the emission source (see Dulk, 1985). The MW gyrosynchrotron emission is
thought to be more sensitive to accelerated electrons than hard X-rays. At
lower frequencies, the emission is usually associated with a two-stages
coherent plasma mechanism. Electron beams with energies of tens of keV become
unstable to Langmuir wave generation. Radio waves are then produced via the
nonlinear transformation of these Langmuir waves into electromagnetic emission
at frequencies around the local plasma frequency and its second harmonic (see
Reid and Ratcliffe, 2014, as a review).

The existence of temporal correlations between coherent and incoherent flare
radio emissions has been known since earlier studies (e.g. Neylan, 1959),
where type III bursts were found to be generally accompanied by enhanced
microwave and hard X-ray flux. Assuming that both types of emission are
induced by the same population of accelerated electrons, analysis of the time
delays between them allows, in particular, for detecting the direction of the
electron beam propagation and estimating its speed. For example, the temporal
connection between type III spikes, hard X-ray spikes, and microwaves were
studied by Dennis et al. (1984); Aschwanden et al. (1995); Reid, Vilmer, and
Kontar (2014); Kupriyanova et al. (2016).

An important and perhaps intrinsic feature of solar flares is quasi-periodic
pulsations (QPPs), frequently seen in the time profiles of flare emission
across the entire electromagnetic spectrum. Statistical studies have shown
that QPPs are present in both non-thermal (Kupriyanova et al., 2010) and
thermal (Simões, Hudson, and Fletcher, 2015) emissions of more than 80% of the
analysed solar flares (see also Hayes et al., 2020, for a more recent
statistics of QPP events). QPPs are usually considered as manifestations of
magnetohydrodynamic (MHD) wave processes in various magnetic plasma
configurations in or nearby the flaring site. Simultaneous analysis of QPP
properties in different spectral bands, such as their timescales, amplitudes,
and phases, opens up new and promising opportunities for studying the
processes of energy release and energy transport throughout the solar
atmosphere (see Kupriyanova et al., 2020; Zimovets et al., 2021, for the most
recent reviews).

Signatures of QPPs are regularly observed in coherent and incoherent radio
emissions. For example, Mészárosová et al. (2009) and Mészárosová, Karlický,
and Rybák (2011) detected QPP signals with periods about 71 s and 81 s in the
type IV bursts at 1.1–4.5 GHz, which were associated with the modulation of
the gyrosynchrotron emission by a packet of fast magnetoacoustic waves
propagating in an open plasma structure. The magnetic field in the vicinity of
the observed radio emission sources was estimated around 11 G and 48 G. Kumar,
Nakariakov, and Cho (2017) observed similar $\sim$70-s and $\sim$140-s quasi-
periodicity in the type IV and type III radio bursts at 245–610 MHz and in
soft X-rays. Moreover, these radio-band QPPs were found to occur
simultaneously with a coronal fast magnetoacoustic wave train, directly
observed in Extreme Ultraviolet (EUV) images. Likewise, signatures of standing
fast magnetoacoustic waves of a sausage symmetry with periods between $\sim$1
s and $\sim$15 s were found as QPP signals in coherent (e.g. Nakariakov et
al., 2018) and incoherent (e.g. Kolotkov et al., 2015) radio emissions from
weak (B-class) and strong (X-class) solar flares, respectively. For more
details on the dynamics of fast sausage waves in the corona and their role in
modulation of the flare emission in the radio band, see the most recent review
by Li et al. (2020).

Higher up in the corona, at radio frequencies 42–83 MHz, Goddard et al. (2016)
detected quasi-periodic “sparks” in the type II burst, whose period in the
radio band (1.78 min) matches the period of a quasi-periodic fast-propagating
wave train observed in images of the lower corona in the EUV band. Moreover,
the speed of the emission location of the observed radio sparks, estimated
from the dynamic radio spectrum, was found to be comparable to the speed of
the coronal mass ejection, inferred from the EUV data. Kupriyanova et al.
(2016) found QPPs present in MWs, hard X-rays and type III bursts with a
common period of 40–50 s. Their findings indicated that the detected
periodicity was probably associated with periodic dynamics in the injection of
non-thermal electrons, which can be produced by periodic modulation of
magnetic reconnection. Another example of the modulation of the low-frequency
radio emission by the fast MHD wave was found by Kolotkov, Nakariakov, and
Kontar (2018) as quasi-periodic drifting fine structures in a type III radio
burst at 33–40 MHz, using data from the LOw Frequency ARray (LOFAR). The
observed wave period of $\sim$3 s and apparent wave propagation speed were
used to probe the coronal magnetic field at $\sim$1.7 solar radii and compare
the energy flux carried by the wave to local radiative losses.

In this work, we aim to reveal the role of MHD wave processes in the energy
release and transport in solar flares. Our study develops the idea of Benz,
Battaglia, and Vilmer (2011) that the electrons may get accelerated not in the
site of primary energy releases but also due to local interactions of the
magnetic plasma structures residing in the active region. For this purpose, we
applied the analysis of QPP signals simultaneously detected in the MW and
decimetric emission from a C-class solar flare SOL2017-09-05T07:04 as a tool
for studying possible correlations and interplay between the mechanisms
producing both types of emission.

The paper is organised as follows. The description of the spatially,
temporally, and spectrally resolved observational data and their analysis are
given in Section s:Observations. The detected properties of the MW and
decimetric QPP events are described in Section s:qpp. The discussion of the
obtained observational results and future perspectives are outlined in Section
s:DiscussionC.

##  2 Observations: data and analysis

s:Observations

The analysed event occurred in the active region (AR) 12673, which produced a
series of powerful flares in September 2017. The event started at around 07:04
UT 05 September 2017, approximately 25 min after the M3.6 GOES class solar
flare, and had a GOES class of C6.9.

###  2.1 Siberian Radioheliograph-48

s:Data_SRH

Our study of the microwave flare emission is based on the data from the
Siberian Radioheliograph-48 (SRH-48, Lesovoi et al., 2014, 2017). In 2016, the
prototype of a T-shaped 48-antenna array (SRH-48) started observations within
the 4–8 GHz frequency band. The instrument uses an imaging approach based on
the Fourier synthesis (Lesovoi et al., 2017). Both circularly-polarised
components ($R$ and $L$) are measured. SRH-48 observed at five frequencies —
4.5, 5.2, 6.0, 6.8, and 7.5 GHz from July 2016 to May 2018. In the current
study, we used only intensity ($R+L$) values. The bandwidth of the frequencies
is 10 MHz. The switch time between the frequencies was about 2 s, while the
accumulation time at each frequency was 0.28 s. The best spatial resolution
was 70”, achieved at 8 GHz. The image processing was done by the original
software developed and tested to produce raw solar images, clean and calibrate
them in brightness temperature. The method of image calibration in brightness
temperatures is described by Kochanov et al. (2013). We note that the obtained
brightness temperatures of compact sources can be considerably lower than the
original temperatures if the source is less than the beam pattern of SRH-48.

For a study of temporal evolution, we used so-called correlation plots. They
represent a proxy of radio ﬂux and display a temporal variation of the sum of
cross-correlations of all antenna pairs. This type of data is very convenient
for solar activity monitoring and are characterised by very high sensitivity
to variations both in the brightness of compact sources and in their
structure. Real-time correlation plots and quick-look images produced by SRH
at ﬁve frequencies are accessible online at
http://badary.iszf.irk.ru/srhCorrPlot.php. Methods for calculating the
correlation plots and their relation with characteristics of solar emission
are presented in Lesovoi and Kobets (2017). The authors showed that the
sensitivity of SRH-48 to the emission of compact sources was up to 10-2
s.f.u., and that 1% increase in the correlation corresponds to an increment of
about 5–10 s.f.u. within the 4–8 GHz band.

###  2.2 Mingantu Spectral Radioheliograph

s:Data_MUSER The Mingantu Spectral Radioheliograph (MUSER) observes the Sun
with high temporal, spatial and spectral resolutions in a wide range of
frequencies (Yan, Chen, and Yu, 2016; Yan et al., 2021). It consists of two
instruments: MUSER-I and MUSER-II. The spectral band of MUSER-I is 0.4–2 GHz.
It has a spatial resolution of 51.6–10.3” and 25 ms temporal resolution.
MUSER-II observes within 2–15 GHz with 10.3–1.3” spatial resolution and 206.25
ms temporal resolution. The data are processed using the original software,
including calibration and image processing (Yan et al., 2002; Wang et al.,
2013; Tan et al., 2015). In total, MUSER has obtained more than 390 TB of
observational data during 2014–2019. There are 85 solar radio burst events,
and among them, more than 60 radio burst events contain fine structure. In the
current study, we used the dynamic spectrum obtained by MUSER-I with high
spectral and temporal resolution.

###  2.3 Microwave spectral data

s:Data_sp Analysing oscillations in solar flare emission, we must be sure that
the detected periods are not instrumental. To ensure their solar origin, we
analysed the same spectral band using independent instruments. The first
instrument is the BroadBand Microwave Spectropolarimeters (BBMS), observing
within 3.8–8.2 GHz (Zhdanov and Zandanov, 2011, 2015). BBMS covers this
frequency range using 16 channels with a cadence of 0.6 s. The second
instrument providing observations of the solar radio flux at similar
frequencies is the Radio Solar Telescope Network (RSTN). We used data from the
Learmonth station obtained within 245 MHz and 15.4 GHz with a 1-s cadence
(see, for example, Kennewell and Cornelius, 1983).

###  2.4 EUV and X-ray observations

s:Data_EUV

Due to an eclipse of the Solar Dynamics Observatory, the analysed solar flare
SOL2017-09-05T07:04 was only observed in the EUV band by the Sun Watcher with
Active Pixel System detector and Image Processing (SWAP) telescope onboard the
ESA PROBA2 (Seaton et al., 2013; Halain et al., 2013). We used SWAP images
obtained in the 174 Å band with a spatial resolution of 3.16” per pixel.

In the X-ray band, the event was observed by GOES only. We used time profiles
of the emission within the 1–8 Å band that is sensitive to thermal processes,
and the derivative of this time profile for studying the behaviour of
accelerated electrons.

###  2.5 Event description and analysis of spectral and spatial features of
the flare sources

Figure 1.: Comparison of microwave, decimeter radio and X-ray emission time
profiles. The left panel: The evolution of emission at 600 MHz by MUSER and
RSTN observations vs soft X-ray time profile within 0.5–4 Å band by GOES and
its derivative. The right panel: The evolution of the MW flux observed by
SRH-48, BBMS, and RSTN at about 8 GHz vs the derivative of the soft X-ray time
profile within 0.5–4 Å by GOES.fig:GOES

The comparison of the radio observations to the X-ray data is shown in Figure
fig:GOES. We present the time profiles in the decimeter radio and MW bands
from several independent instruments; the original SRH-48 correlation plot,
BBMS, MUSER, and RSTN. The cross-instrumental analysis shows a close
correlation between the time profiles obtained by different remote instruments
at similar frequencies. This fact allows us to conclude that the event and its
fine structure had a solar origin and was not related to instrumental or
terrestrial ionospheric effects. As we can see in Figure fig:GOES, both bursts
occurred during the rise phase of the flare.

We analysed the derivative of the soft X-ray time profile (SXR) in the 0.5–4 Å
band for the role of accelerated electrons in the generation of the analysed
emissions (Neupert effect, Neupert, 1968). One can note a close agreement of
both MW and decimeter radio bursts with the derivative of the SXR time
profile. Moreover, we see that the 600 MHz burst was shorter in duration and
occurred at the maximum of the time profile of the SXR derivative. These facts
indicate the presence of accelerated electrons and their possible contribution
to the generation of radio emissions. Comparison of the decaying phase of the
MW and the SXR derivative time profiles demonstrates the absence of the trap
effect (see, for example, Melnikov, 1994). Thus, we can exclude the magnetic
trap from the flare topology and the model “trap plus precipitation” from
interpretation (see, Silva, Wang, and Gary, 2000).

Figure 2.: The left panel: Radio spectra with two maxima around 600 MHz and
6.6 GHz, obtained from the data by BBMS, MUSER, and RSTN. The blue colour
shows flux at 07:05:46 UT (peak of the 600 MHz flux by RSTN). The red colour
marks data at about 07:05:55 UT (peak of the MW flux). The black line is
$f^{2.5}$ function. The right panel: The SWAP 174 Å image with the overlaid MW
sources obtained by SRH-48. The sources are defined as the difference between
the image at 07:05 UT and the image at 07:00 UT when flare activity in MW was
absent. fig:EUV

We constructed the radio spectra from 100 MHz to 20 GHz using data from MUSER,
BBMS, and RSTN for two instants in time. The first moment was chosen at the
maximum of the 600 MHz time profile by RSTN. The left panel of Figure fig:EUV
shows this spectrum in blue. The second moment was chosen at the maximum of
the MW burst, with the spectrum shown in red in Figure fig:EUV. Both MW
spectra have a typical gyrosynchrotron form. The spectrum nearby 600 MHz is in
turn very narrow-band and shows the form characteristic for the plasma
emission mechanism.

We obtained the peak frequency of the MW spectrum (the frequency of the
maximum flux value) and the high-frequency photon spectral index using the
technique by Nita, Gary, and Lee (2004). The peak frequency is found to be
about 6.6 GHz for both moments of time considered. Thus, the SRH data at 6.8
GHz and 7.5 GHz corresponds to the optically thin part of the spectrum and
indicates the segments of the flare loop near the footpoints. The MW sources
at 4.5 GHz and 5.2 GHz in turn correspond to the optically thick emission from
the entire flare loop. The photon spectral indices at high MW frequencies are
found to be about 2.9 and 2.5 for the first and second moments, respectively.
These values are estimates for the non-thermal electron spectral power-law
index. The spectral index at low MW frequencies is about 2.5 (see Figure
fig:EUV), indicating an absence of the Razin effect and a normal (not
anomalously high) density within the emitting loop (see, for example, Dulk,
1985).

AR 12673 is well-known for strong magnetic fields and well-developed magnetic
structures, which resulted in a series of powerful solar flares (see e.g. Wang
et al., 2018; Kudriavtseva et al., 2021). To exclude the contribution of
cyclotron emission from the sunspots and other permanent MW sources, we
subtracted the image obtained at 07:00 UT (i.e. before the analysed flare)
from the images obtained during the MW burst. The MW sources which appeared at
07:05 UT are overlaid as colour contours on a background 174 Å image by SWAP
(see the right panel in Figure fig:EUV). The 6.8 GHz and 7.5 GHz images
indicate a compact source coming from the vicinity of a flare loop footpoint,
while the images at lower frequencies 4.5 GHz and 5.2 GHz demonstrate the
entire flare loop structure. The distance between the centroids of 4.5 GHz and
7.5 GHz sources is about 43” or 31 Mm (assuming 1” is about 725 km). Assuming
the flare loop geometry is a half-circle, we can consider this distance
between the centroids as the loop height. This value agrees with the size of
an average flare loop usually seen in X-rays. Moreover, the previous studies
of decimetric pulsation positions relative to coronal X-ray source position
revealed that they are spatially separated by hundreds of arcseconds. This
fact indicates that the radio burst emission was not generated by the
electrons trapped in the flare loop, but likely came from a current sheet
situated well above the X-ray loop top (Battaglia and Benz, 2009; Benz,
Battaglia, and Vilmer, 2011).

We note that the sources at 6.8 GHz and 7.5 GHz are seen in Figure fig:EUV to
situate near the fan-like structure clearly visible in the 174 Å image by SWAP
(marked by the magenta arrow in Figure fig:EUV, right panel). This structure
was apparently long-lived and also visible in 171 Å images of SDO/AIA before
and after the time interval analysed in this work.

Figure 3.: The MUSER dynamic spectra of the intensity $R+L$ (left) and
polarisation $R-L$ (right). Colour bars indicate the corresponding fluxes in
s.f.u. The horizontal line in the left panel indicates the frequency where the
dynamic spectrum reaches its maximum value.fig:DynSpectrum

Figure fig:DynSpectrum presents the dynamic spectrum by MUSER both for
intensity ($R+L$) and Stokes parameter $V$ or circular polarisation ($R-L$).
The intensity spectrum on the left panel shows a series of narrow-band, quasi-
periodic bursts similar to the type I bursts, which are known to have narrow
bandwidth and short duration (see, for example, Kai, Melrose, and Suzuki,
1985). Classical type I noise storms are usually observed below 400–450 MHz,
corresponding to coronal altitudes around 0.2 solar radii (Mercier et al.,
2015). We note that type I radio bursts are associated with non-thermal
electron emission. This fact allows us to assume a similar emission mechanism
for the considered decimetric bursts.

The decimeter bursts are polarised, as one can see in the right panel of
Figure fig:DynSpectrum presenting the Stokes $V$ parameter. The presence of
polarisation indicates the non-thermal origin of the emission. The sign of
polarisation does not change with time. It is mostly positive, but we can see
the sign inverts at frequencies below $\approx$ 600 MHz. This inversion of the
polarisation sign indicates that the emission at those frequencies could
originate near the top of a high loop (substantially higher the one observed
in MW in Figure fig:EUV).

##  3 Quasi-periodic properties of the microwave and decimeter emission

s:qpp The microwave emission (Figure fig:GOES) shows quasi-periodic behaviour
with the period around 20–30 s during the flux rise phase from 07:04:10 UT to
07:06:30 UT. The dynamic spectra shown in Figure fig:DynSpectrum also
demonstrate quasi-periodic enhancements of the decimeter emission which is
clearly seen in the intensity from 07:05:22 UT to 07:06:05 UT, i.e. around
maximum of the microwave flux (at 07:05:40 UT), with a shorter period. In this
section, we perform a detailed analysis of the periodic properties of the MW
and decimeter intensities shown in Figures fig:GOES and fig:DynSpectrum. The
technique which we apply for the detection of QPPs in the MW and decimetric
time profiles consists of removing their low-frequency trends, and processing
the detrended time series using decomposition techniques.

The problem of detrending in the context of QPP detection, especially in the
presence of coloured noise, is described in, for example, Kolotkov et al.
(2018). Different techniques currently used for the detection and analysis of
QPP signals in solar flares were summarised and comprehensively tested in
Broomhall et al. (2019), see also Kupriyanova et al. (2020). In our study, we
determine the low-frequency trends of the observed emissions by smoothing the
original time profiles through the convolution with a polynomial function
(also known as a Savitzky–Golay filter), applied with a smoothing time window
of the width $\tau$. We show this width in Figure fig:Wavelet_BBMS and Figure
fig:Wavelet_MUSER by the horizontal dotted lines. After subtracting the trend,
the detrended time series were analysed with both the wavelet (Morlet)
transform (Torrence and Compo, 1998) and the Fourier periodogram. The
statistical significance level in the wavelet analysis was obtained for the
red noise function. The significance level for the Fourier periodogram is
calculated according to Horne and Baliunas (1986). Such a combination of the
Fourier and wavelet approaches accompanied by a rigorous assessment of the
statistical significance of the obtained periodic components in comparison
with the background noise allows us to exclude possible artifacts of the data
processing.

###  3.1 QPPs at microwaves

s:qpp_MW

Figure 4.: Top panel: the BBMS flux at 8.056 GHz (black curve) and its low-
frequency trend (blue curve). Bottom left: the Morlet wavelet power spectrum
(gradient of red) of the de-trended flux component. Bottom right: the
corresponding global wavelet (black line) and Fourier (red line) spectra. The
significance level is shown by the green contour and the dashed line in the
bottom left and right panels, correspondingly. The horizontal lines indicate
the width $\tau=50$ s of the smoothing window used for obtaining the trend.
The vertical dashed lines show the time interval 07:05:15–07:06:15 UT where
the decimetric QPPs were found (see Figure fig:Wavelet_MUSER).fig:Wavelet˙BBMS

An example of the original time profile of the BBMS flux at 8.056 GHz, its
low-frequency trend obtained for $\tau=50$ s, and the corresponding wavelet
and Fourier power spectra calculated after removing the trend are shown in
Figure fig:Wavelet_BBMS. As such, the wavelet analysis detects a single-mode
QPP signal with the characteristic period around $30\pm 10$ s. The Fourier
periodogram of the auto-correlation function of the detrended signal reveals
the same period $30\pm 5$ s.

Figure 5.: Comparison of the de-trended oscillation time profiles at different
frequencies: modulation depths of the fluxes in the microwave (left panel) and
decimeter (right panel) bands. fig:Detrended

These quasi-periodic variations with the same period about 30 s are detected
in fluxes at all microwave frequencies within the 4–8 GHz band. Moreover, the
QPPs for all these time profiles demonstrate an in-phase behaviour, as shown
by Figure fig:Detrended. Here, the modulation depth, which is defined as the
ratio of the detrended flare light curve to its low-frequency trend, is
illustrated for three frequencies; the optically thick part of the MW spectrum
(4.545 GHz, see the left panel in Figure fig:EUV), near the spectral maximum
(6.732 GHz), and in the optically thin part of the spectrum (8.056 GHz). The
modulation depths are found to be around a few percent for all the profiles.
The local differences are probably caused by noise in the raw data and time
binning.

###  3.2 QPPs at decimeters

s:qpp_dm

Figure 6.: The same as in Figure fig:Wavelet_BBMS but for the flux registered
by MUSER at 675 MHz.fig:Wavelet˙MUSER

A QPP signal with a shorter period of $6\pm 1.5$ s and at least seven well-
pronounced oscillation cycles is found in the time profiles obtained with
MUSER at all frequencies from 500 MHz to 700 MHz. This period is well seen in
the original time profiles, wavelet spectra, and Fourier periodograms, as
demonstrated in Figure fig:Wavelet_MUSER for the radio flux intensity at 675
MHz, where the amplitude of the flux and its variations reach their maxima
(see Figure fig:DynSpectrum). The results are the same for different smoothing
widths $5$ s $<\tau<20$ s, while the flux variations with longer timescales
are associated with the relaxation of the local plasma conditions after a
primary energy release. For all radio frequencies, the 6 s QPPs are dominant
and statistically significant in the wavelet spectrum. Similarly to the QPPs
in microwaves, all the QPPs in the decimeter range are in-phase, with a
modulation depth around 100% (see Figure fig:DynSpectrum and Figure
fig:Detrended).

##  4 Discussion and conclusion

s:DiscussionC

We performed analysis of quasi-periodic pulsations (QPP) simultaneously
observed in the microwave and decimetric emissions of SOL2017-09-05T07:04 C6.9
solar flare to explore and establish the relationship between these two radio
emission mechanisms. More specifically, we found the following properties of
the QPP events in microwaves:

  * •

Three clear cycles with an oscillation period of about 30 s.

  * •

Both the peak frequency of the microwave spectrum and photon spectral index
did not change during the QPPs.

  * •

The spectral index at low frequencies indicates the absence of the Razin
effect.

  * •

Time profiles of the MW oscillations demonstrate an in-phase behaviour both in
the optically thick and optically thin emissions.

  * •

The modulation depth of the oscillations is similar at different frequencies,
and its values are about a few percent.

and in decimeters:

  * •

At least seven oscillation cycles with an oscillation period of about 6 s.

  * •

The frequency of the maximum intensity is about 675 MHz, and this did not
change during the event.

  * •

The QPPs appear in the narrow spectral band 500–700 MHz.

  * •

The sign of the Stokes $V$ parameter did not change in time during the event,
but it reversed at the lowest frequency.

  * •

Time profiles of the decimeter oscillations demonstrate an in-phase behaviour.

  * •

The modulation depth of the oscillations is around 100%.

Figure 7.: A schematic illustration of a spatial structure of the loop
boundary perturbed by a fundamental harmonic ($N=1$), third harmonic ($N=3$),
and their superposition ($N=1+3$) of a standing fast sausage wave. The
perturbations near the loop legs, caused by the third sausage harmonic, allow
for the periodic interaction with a nearby plasma structure and acceleration
of charged particles. In our scenario, the external plasma structure is
considered to be a closed magnetic loop, substantially higher than the
sausage-oscillating MW loop. The thin black lines show the unperturbed state
of the loop. The choice of the colour scheme is arbitrary.fig:sketch

The observed oscillation period $P_{1}=30$ s and an in-phase behaviour of the
QPP signals in the MW band are in good agreement with the modulation of the
gyrosynchrotron emission by a sausage mode of a standing fast magnetoacoustic
wave in the coronal loop (see e.g. Melnikov et al., 2005; Inglis, Nakariakov,
and Melnikov, 2008; Kuznetsov, Van Doorsselaere, and Reznikova, 2015, for
observational and modelling examples of this scenario). Moreover, impulsive
energy releases could naturally excite several harmonics of the sausage mode
simultaneously in the coronal loop. The first and the most powerful harmonic
of the sausage mode is known to have nodes near the loop legs and anti-nodes
(i.e. maximum perturbation amplitude) at the top of the flaring loop (see
Figure fig:sketch, for a schematic illustration). Likewise, the MW emission at
the frequencies around the MW spectral peak and below are also known to be
stronger near the loop top (see e.g. Figure 1 in Bastian, Benz, and Gary,
1998). Hence, the fundamental sausage harmonic should dominate at these MW
frequencies, as it is observed at 4–8 GHz in this work. No signatures of
higher sausage harmonics were detected in the MW band.

As we demonstrated in Section s:Observations, the flare loop seen in MW is far
too low to emit in the decimetre range. Hence, we are not able to use a
single-loop model for the interpretation of the 6 s QPP signal at 500–700 MHz.
On the other hand, the MW flare loop is seen in Figure fig:EUV to situate
nearby an apparently open or high loop structure seen in the 174 Å EUV
channel. More specifically, the lower part of the MW loop, observed at 6.8 and
7.5 GHz, is seen to be closer to this external loop-like structure than the
loop top. We note here that the MW sources at 6.8 and 7.5 GHz indicate not the
footpoint of the flare loop but a segment above it. Thus, we suggest that the
observed 6 s decimetric QPPs could be attributed to the effect of a higher
sausage harmonic of the flare loop, which has non-zero perturbation amplitudes
near the loop legs, thus allowing for an effective periodic interaction of the
sausage-oscillating loop with this external plasma structure and formation of
magnetic X-points and current sheets, as schematically illustrated in Figure
fig:sketch.

For example, the third harmonic of a sausage-oscillating flare loop is
expected to have a period of $P_{3}=P_{1}/3\approx 10$ s (or $\gtrsim 10$ s
taking the effect of waveguiding dispersion into account, see e.g. Inglis et
al., 2009; Yu et al., 2016, for details). Following a qualitative scenario
developed by Nakariakov et al. (2006), this essentially transverse very small-
amplitude wave modulates the plasma conditions at the interface between the
sausage-oscillating loop and the external loop, causing, in particular, strong
variations of the local electric current density with the period of the mother
higher-harmonic sausage oscillation, i.e. around 10 s. For the local current
density periodically exceeding some threshold value, the plasma is known to be
subject to various micro-instabilities and micro-turbulence, which can in turn
lead to anomalous values of the plasma resistivity and thus provide plausible
conditions for periodic triggering of magnetic reconnection, acceleration of
charged particles and hence bursts of low-frequency coherent radio emission.
Moreover, as the extreme values of the current density are achieved at both
positive and negative half-cycles of the oscillation (see Figure 3 in
Nakariakov et al., 2006), the rate at which magnetic reconnection triggers in
this scenario is two times higher than the 10 s period of the higher-harmonic
sausage wave inducing it. In other words, the third harmonic of the sausage-
oscillating loop with a 10 s period is expected to trigger magnetic
reconnection that can cause the acceleration of charged particles every 5 s in
this scenario, which is very close (or identical within the uncertainties that
we have in the data analysis) to the observed 6 s period in the decimetric
band.

It is also worth mentioning here that in addition to perturbing the loop near
the loop legs where the interaction with the external plasma structure
potentially takes place, the third sausage harmonic has a non-zero
perturbation amplitude near the loop top where the maximum of the MW emission
at 4–8 GHz is observed. However, we detected no signatures of this higher
harmonic in the MW observations, mainly due to its low amplitude in comparison
with the fundamental sausage harmonic dominating in this part of the loop. In
any case, the suggested interpretation should be considered as a hypothesis
only, as it is very difficult to find the direct evidence of reconnection in
observations. On the one hand, it is caused by the lack of the EUV data
available for this event. On the other hand, previous studies demonstrated
that small-scale quasi-periodic reconnection does not necessary change the
global magnetic topology of the flare region (see e.g. Kliem, Karlický, and
Benz, 2000; Murray, van Driel-Gesztelyi, and Baker, 2009). We also note that
the propagating fast magnetoacoustic waves cannot be completely ruled out in
this interpretation. For example, the detected form of the wavelet power
spectrum shown in Figure fig:Wavelet_BBMS resembles a characteristic tadpole
structure typical for quasi-periodic fast-propagating wave trains (see e.g.
Nakariakov et al., 2004; Mészárosová et al., 2014; Kolotkov et al., 2021). On
the other hand, it is less clear how to explain the observed ratio of the
oscillation periods in MW and decimeters by propagating waves.

The oscillation modulation depth found in the microwave emission does not
exceed 2–3$\%$, while the oscillation modulation depth detected in the
decimeter band is about 100$\%$. The modulation depth found in the microwave
emission agrees with the results obtained by modelling of the microwave
emission from a sausage-oscillating loop, which showed the modulation depth
around 1–5$\%$ (Kuznetsov, Van Doorsselaere, and Reznikova, 2015). The
observed disparity in the microwave and decimeter modulation depths is likely
to occur due to the difference in the emission generation mechanisms. Namely,
the microwave emission is dependent upon many parameters such as accelerated
electron flux, magnetic field, background plasma density, source size and
flare loop inclination (see Dulk, 1985, for details), which for some
combinations could lead to the observed relatively low modulation amplitudes.
For the coherent plasma emission, this is likely stimulated by deca-keV
electrons in a much larger loop structure. Each burst of radio emission is
potentially caused by a separate acceleration event, creating the high
modulation depth of 100%.

To the best of knowledge, this work presents the first collaborative analysis
of the data from SRH and MUSER. It allowed us to establish the relationship
between the oscillations detected in the MW and decimetric emission of the
solar flare. The scenario explaining the obtained periodicities is similar to
that considered by Benz, Battaglia, and Vilmer (2011), with the magnetic
topology of the event assuming the current sheet that generates the decimetric
emission is located not above the loop-top source but nearby the flare loop.
Clearly, the proposed physical model is rather qualitative than quantitative
and needs further validation by using comprehensive theoretical simulations
and forward modelling, addressing, in particular, a huge disparity in the
observed QPP modulation depths in the MW and decimetric bands. Despite this,
the work demonstrates a clear potential of multi-band radio observations for
deeper understanding of the mechanisms of particle acceleration and
propagation during solar flares and the role of MHD waves in these processes.

#### Acknowledgments

This study was funded by the RFBR (project number 21-52-10012) and the Royal
Society (project number IEC\R2\202175). The work of LK, AK and DK was partly
financially supported by the Ministry of Science and Higher Education of the
Russian Federation. The work of EK and DK on the detection of quasi-periodic
signals and interpretation of results was supported by Russian Scientific
Foundation grant 21-12-00195. DK was also supported by the STFC consolidated
grant ST/T000252/1. CT was was supported by the NSFC grants 11790301 and
11941003. Authors thank CSUC ’Angara’ for BBMS data, teams operating SRH-48,
MUSER, PROBA2/SWAP, RSTN and GOES for open access to observational data.

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