1 Introduction

Solar flares are explosive phenomena in the solar atmosphere in which the energy accumulated in the magnetic field and electric currents is rapidly transformed into plasma heating, plasma flows, accelerated particles, and emissions in a broad range of electromagnetic waves: from radio to optical, UV, X-rays, and to gamma rays (Priest and Forbes, 2002; Krucker et al., 2008; Schrijver, 2009; Fletcher et al., 2011; Aulanier, Janvier, and Schmieder, 2012; Nakariakov et al., 2016).

In the radio range, solar flares are associated with many types of radio bursts, see books by Krueger (1979), McLean and Labrum (1985), Aschwanden (2004). Besides the type III bursts produced by particle beams, type II bursts produced by shocks, and type IV bursts with their fine structures like narrowband spikes, zebras, and fibers (Chernov, 2011), there are drifting pulsation structures (Kliem, Karlický, and Benz, 2000) and lace bursts (Karlický et al., 2001).

Type IV bursts are also denoted as continuum bursts. For their subclasses, see Figure III.26 in the book by Krueger (1979). Among these subclasses, there are the continua observed in the decimetric range. They are usually recorded during long-lasting flares after their impulsive phase (denoted also as type IV dm).

Radio bursts are characterized by time and frequency variations. Applying the power spectrum analysis (PSA), these variations are used in a search for oscillatory and turbulence processes. As far as turbulence is concerned, this PSA was exploited for narrowband dm-spikes first (Karlický, Sobotka, and Jiřička, 1996; Karlický, Jiřička, and Sobotka, 2000). Later, the fine structure of type IIIb bursts (Chen et al., 2018) was analyzed. In both cases, a power spectrum of burst intensity variations with a spectral index close to −5/3 was found. Recently, Carley et al. (2021) studied intensity variations of herringbones in the type II bursts using PSA and derived the power-law index in the −1.7 – −2.0 range. Besides the studies analyzing burst intensity variations, PSA was also used in the analysis of frequency variations of zebra stripes (Karlický, 2014), where the power-law index was also close to −5/3.

In the present paper, we study frequency variations of the frequency boundaries of decimetric continua for the first time. Since decimetric continua are relatively frequently observed, their analysis by PSA method can be used in studies of turbulence in solar flares.

2 Observations and Their Analysis

Looking at the 800 – 2000 MHz radio spectra observed in the period 2004 – 2022 by the Ondřejov radiospectrograph (Jiřička and Karlický, 2008) we selected five decimetric continua that have distinct frequency boundaries and had a duration longer than 6 minutes. Then, in all these continua we took parts in which frequency boundaries were continuous and without any disruption by artificial sources. To be above spectral noise in these cases, a time resolution of 0.6 seconds was taken. The analyzed continua, their boundary, and corresponding GOES solar flares are summarized in Table 1. Only in the 25 June 2015 continuum its high- and low-frequency boundary were within the spectral range of the radio spectrum (800 – 2000 MHz). In other cases, the low-frequency boundary was lower than 800 MHz. The global frequency drift of these continua was close to zero. Except for the 30 December 2004 continuum, where no fine structure was observed, fiber bursts were found in some short time intervals during all other continua. Moreover, in the 1 August 2010 continuum in some short time intervals during continuum we observed zebras; see Karlický (2014).

Table 1 Analyzed intervals of decimetric continua, their boundary, and corresponding GOES solar flares according to NOAA reports. H or L denote the high- or low-frequency continuum boundary.
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Let us start with analysis of the 25 June 2015 continuum, where both continuum boundaries were in the 800 – 2000 MHz range. In its radio spectrum (Figure 1), we selected a radio flux level and for this level we obtained the time-frequency profile of the continuum boundaries: see Figure 1A for the high-frequency boundary and Figure 2A for the low-frequency one. These profiles correspond to the narrow time varying gray-olive bands in the radio spectrum (Figure 1). While the selected time–frequency profile for the high-frequency boundary lasts 13 minutes (09:11 – 09:24 UT), the time interval for the low-frequency boundary was taken only for 6 minutes (09:11 – 09:17 UT), due to artificial noise near 960 MHz.

Figure 1
figure 1

Upper part: Radio spectrum of the decimetric continuum observed during the 25 June 2015 flare. Bottom part: A) The time-frequency profile of the high-frequency continuum boundary that corresponds to the narrow gray-olive band in the radio spectrum in the 09:11-09:24 UT interval. B) Power spectrum of the frequency profile (FRE) shown in A). C) Radio flux in the same time interval as in A) at 1478 MHz, and D) corresponding power spectrum (RFLUX). Dashed lines with the power index -5/3 in B) and D) are shown for comparison.

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Figure 2
figure 2

A) Time-frequency profile of the low-frequency continuum boundary that corresponds to the narrow gray-olive band in the radio spectrum (Figure 1 upper part) in the 09:11-09:17 UT interval. B) Power spectrum of the frequency profile (FRE) shown in A). C) Radio flux in the same radio spectrum (Figure 1) and same time interval as in A) at 1050 MHz, and D) corresponding power spectrum (RFLUX).

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Then, the time–frequency profiles were analyzed by the Fourier method and the power spectra of frequency variations of continuum boundaries were obtained: see Figure 1B for the high-frequency boundary and Figure 2B for the low-frequency one. Both spectra have a power-law form and the power-law index is close to −5/3, as shown by comparison with the dashed lines in these figures. The radio flux level used to determine the time-frequency profiles was taken arbitrarily over a range, but tests have shown that changes of the flux level do not change the power-law index.

Then, we studied radio flux variations at frequencies that are in the range of varying boundary frequencies. Since these variations in this range were highly correlated with the frequency boundary variations, we calculated their cross-correlation coefficients. Thus, for the high-frequency boundary, we took the radio flux variations at the frequency with the maximal cross-correlation with the boundary frequency variations, see Figure 1C and its power spectrum (Figure 1D). For the low-frequency boundary, we took the radio flux variations at the frequency with the minimal cross-correlation, see Figure 2C and also its power spectrum (Figure 2D). As shown in Table 2, the cross-correlation coefficient for the high-frequency boundary is 0.92 and for the low-frequency boundary −0.87. The lag in the cross-correlation calculations was 0 seconds. In both power spectra (Figures 1D and 2D), the power-law index is close to −5/3.

Table 2 Cross-correlation coefficients and their lags between the frequency boundary variations and radio flux shown in plots A) and C) in Figures 1 – 6. H or L denote the high- or low-frequency continuum boundary.
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In other continua (Table 1), we studied only variations at the high-frequency boundary. The procedure was the same as in the 25 June 2015 continuum. The corresponding results are shown in Figures 3 – 6. In one case (24 February 2011), for comparison, we added the hard X-ray emission profile, observed by FERMI monitor (Meegan et al., 2009); see the dotted line in Figure 5C. As seen in this figure, at some times the correlation between radio flux and hard X-rays is high (around the time 50 seconds, i.e., at about 07:33 UT), otherwise it is low.

Figure 3
figure 3

Upper part: Radio spectrum of the decimetric continuum observed during the 30 December 2004 flare. Bottom part: A) The time-frequency profile of the high-frequency boundary that corresponds to the narrow gray-olive band in the radio spectrum. B) Power spectrum of the frequency profile (FRE) shown in A). C) Radio flux in the same time interval as in A) at 1506 MHz, and D) corresponding power spectrum (RFLUX).

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Figure 4
figure 4

The same as in Figure 3, but for the decimetric continuum observed during the 1 August 2010 flare. Radio flux in C) is at 1497 MHz.

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Figure 5
figure 5

The same as in Figure 3, but for the decimetric continuum observed during the 24 February 2011 flare. Radio flux in C) is at 1384 MHz. The dotted line in C) is the FERMI light curve of the 50–100 keV X-ray emission.

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3 Discussion and Conclusions

In the present paper, we studied five decimetric continua. Their global frequency drift was so small that they look to be stationary. Unfortunately, this cannot be confirmed by positional measurements of their sources. However, it can be supported by observations of fine structures in these continua, because stationary continua are usually rich in fine structures compared with moving continua (Fomichev and Chernov, 2023). In our cases, except for the 30 December 2004 continuum, we found fiber bursts at some short time intervals during all continua. Moreover, in the 1 August 2010 continuum in some short time intervals during continuum, we observed zebras. Thus, we think that the analyzed continua are stationary continua.

As seen in Figures 1 – 6, the power index of the power-law spectra of frequency variations of high- and low-frequency boundaries (FRE) as well as radio flux ones (RFLUX) is close to −5/3. This indicates a turbulence with the Kolmogorov spectral index in decimetric continuum sources. Therefore, we assume that the frequency variations are caused by density variations. Taking radio frequencies equal to plasma frequencies, we recalculated the frequency variations to those of the plasma density. Figure 7A shows the power spectrum of this density variation in the 26 October 2013 continuum. As seen here, the power-law index remains the same as in the power spectrum of the frequency variations, see Figure 6B. For other continua, it is the same.

Figure 6
figure 6

The same as in Figure 3, but for the decimetric continuum observed during the 26 October 2013 flare. Radio flux in C) is at 1173 MHz.

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Figure 7
figure 7

A) Power spectrum of the density profile derived from the frequency in Figure 6, i.e., for the 26 October 2013 continuum. Dashed line with the power index -5/3 is given for comparison. B1) Histogram of boundary frequencies in the 11:19-11:25 UT interval shown in Figure 6 and the corresponding histogram of densities (B2). The Gaussian fit in B2 is indicated by a dotted line. The vertical dashed line shows the mean value of the Gaussian fit. NF and ND mean the number of frequencies and densities per bin size.

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To know the level of the plasma turbulence in the 26 October 2013 continuum, we made a histogram of frequencies as well as a histogram of densities during the 11:19-11:25 UT interval (Figures 7B1 and 7B2). We fitted this density histogram with the Gaussian function and determined the mean value of the density distribution and its half-width as about \(1.66 \times 10^{10}\text{ cm}^{-3}\) and \(1.3 \times 10^{9}\text{ cm}^{-3}\), respectively. Thus, in the 26 October 2013 continuum, the turbulence level is about 7.8% of the mean plasma density.

We made the same calculations for the high- and low-frequency boundaries of the 25 June 2015 continuum (Figure 8). For the high-frequency boundary, we determined the mean value of the density distribution, its half-width, and turbulence level as about \(2.69 \times 10^{10}\text{ cm}^{-3}\) and \(3 \times 10^{9}\text{ cm}^{-3}\) and 11.2%, respectively. On the other hand, for the low-frequency boundary, the mean value of the density distribution, its half-width, and turbulence level is about \(1.31 \times 10^{10}\text{ cm}^{-3}\) and \(1 \times 10^{9}\text{ cm}^{-3}\) and 7.6%, respectively.

Figure 8
figure 8

A1) Histogram of frequencies of the high-frequency boundary in the 09:11-09:24 UT interval of the 25 June 2015 continuum shown in Figure 1A and corresponding histogram of densities (A2). B1) Histogram of frequencies of the low-frequency boundary in the 09:11-09:17 UT interval of the 25 June 2015 continuum shown in Figure 2A and corresponding histogram of densities (B2). The Gaussian fit in A2 and B2 is indicated with a dotted line. The vertical dashed line shows the mean value of the Gaussian fit. NF and ND mean the number of frequencies and densities per bin size.

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In addition to these results on plasma turbulence in sources of dm-continua, we found a high correlation between the frequency variations of high- and low-frequency continuum boundaries and radio flux variations. All the studied continua were observed after the flare impulsive phase, when the X-ray flare emission according to GOES X-ray curves is still enhanced; see also the FERMI light curve of the 50 – 100 keV X-rays in Figure 5C. Therefore, in this flare phase, in agreement with the standard CSHKP flare model (Carmichael, 1964; Sturrock, 1966; Hirayama, 1974; Kopp and Pneuman, 1976), we propose the following explanation of the presented results, see the schematic representation of the dm-continuum source in Figure 9. The plasma outflow from the reconnection site is in a turbulent state. Owing to this outflow, in front of the flare loop the termination shock is formed (Aurass and Mann, 2004; Warmuth, Mann, and Aurass, 2009). The outflow carries the plasma with the magnetic field to the termination shock. Here, electrons are accelerated by adiabatic reflection in a quasi-perpendicular shock. This is known as fast-Fermi acceleration or shock drift acceleration (Holman and Pesses, 1983; Krauss-Varban and Wu, 1989; Vandas and Karlický, 2011). Because the plasma entering the termination shock is in the turbulent state, its magnetic field changes direction and, therefore, the efficiency of the acceleration is changed, as shown by simulations made by Guo and Giacalone (2010). Thus, at the termination shock, spatial–spectral properties of the turbulent plasma outflow are transformed into time–spectral properties of accelerated electron beams. Then these beams propagate upstream or downstream of the termination shock and generate radio emission by the plasma emission mechanism, similarly as in the case of herringbones in the type II bursts. Owing to a very high density gradient in the region of continuum generation, the frequency drift of the beam emission is much higher than in the herringbones case. It is so high that it is not measurable with the used time resolution of radio spectra.

Figure 9
figure 9

Schematic representation of the dm-continuum source.

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We think that this process can explain not only the power-law index −5/3 we found, but also the high correlation between the frequency variations of high- and low-frequency continuum boundaries and radio flux variations. Namely, stronger beams are probably more energetic and thus propagate in a larger interval of heights in the solar atmosphere, i.e., in the wider interval of the plasma densities and broader range of radio frequencies. The beams can have different pitch-angle distributions and propagate through the plasma in turbulent state. Therefore, only some of them reach dense layers of the solar atmosphere, where the hard X-ray emission is produced by the bremsstrahlung mechanism. This could explain that only few peaks in continuum and the 50 – 100 keV X-ray emission of the 24 February 2011 flare are correlated (Figure 5C). Such a low correlation between radio flux and X-ray variations is in agreement with an appearance of fibers and zebras in some short time intervals during continuum observations. Namely, both fine structures need the loss-cone distribution of suprathermal electrons for their generation (Mann, Karlický, and Motschmann, 1987; Benáček and Karlický, 2022). In the assumed turbulent plasma in its local magnetic traps, such a distribution can be formed from the electrons accelerated in the termination shock and then generate observed fibers and zebras. We note that the power-law spectral index of the frequency variations of zebra stripes in the 2010 August 1 continuum is also close to −5/3 (Karlický, 2014), in agreement with the value derived from the frequency variations of the continuum boundaries.

Finally, we think that the used method, applied to frequency variations of decimetric continua boundaries and accompanying fine structures like zebras, can be used in studies of the plasma turbulence in reconnection outflows in a vicinity of the termination shock of solar flares.