Issue 
A&A
Volume 638, June 2020



Article Number  A22  
Number of page(s)  5  
Section  The Sun and the Heliosphere  
DOI  https://doi.org/10.1051/00046361/202037936  
Published online  03 June 2020 
Estimating density and magnetic field turbulence in solar flares using radio zebra observations
^{1}
Astronomical Institute of the Czech Academy of Sciences, Fričova 298, 251 65 Ondřejov, Czech Republic
email: karlicky@asu.cas.cz
^{2}
St.Petersburg State University, 198504 St.Petersburg, Russia
^{3}
St.Petersburg Branch of Special Astrophysical Observatory, 196140 St.Petersburg, Russia
Received:
12
March
2020
Accepted:
10
April
2020
Context. In solar flares the presence of magnetohydrodynamic turbulence is highly probable. However, information about this turbulence, especially the magnetic field turbulence, is still very limited.
Aims. In this paper we present a new method for estimating levels of the density and magnetic field turbulence in time and space during solar flares at positions of radio zebra sources.
Methods. First, considering the doubleplasma resonance model of zebras, we describe a new method for determining the gyroharmonic numbers of zebra stripes based on the assumption that the ratio R = L_{b}/L_{n} (L_{n} and L_{b} are the density and magnetic field scales) is constant in the whole zebra source.
Results. Applying both the method proposed in this work and one from a previous paper for comparison, in the 14 February 1999 zebra event we determined the gyroharmonic numbers of zebra stripes. Then, using the zebrastripe frequencies with these gyroharmonic numbers, we estimated the density and magnetic field in the zebrastripe sources as n = (2.95−4.35) × 10^{10} cm^{−3} and B = 17.2−31.9 G, respectively. Subsequently, assuming that the time variation of the zebrastripe frequencies is caused by the plasma turbulence, we determined the level of the time varying density and magnetic field turbulence in zebrastripe sources as Δn/n_{t} = 0.0112–0.0149 and ΔB/B_{t} = 0.0056–0.0074, respectively. The new method also shows deviations in the observed zebrastripe frequencies from those in the model. We interpret these deviations as being caused by the spatially varying turbulence among zebrastripe sources; i.e., they depend on their gyroharmonic numbers. Comparing the observed and model zebrastripe frequencies at a given time, we estimated the level of this turbulence in the density and magnetic field as Δn/n_{s} = 0.0047 and ΔB/B_{s} = 0.0024. We found that the turbulence levels depending on time and space in the 14 February 1999 zebra event are different. This indicates some anisotropy of the turbulence, probably caused by the magnetic field structure in the zebra source.
Key words: Sun: flares / Sun: radio radiation / turbulence
© ESO 2020
1. Introduction
Solar flares are characterized by fast plasma flows, e.g., by plasma outflows from the magnetic reconnection (Priest & Forbes 2000). If in such plasma flows there are sufficiently strong shears in flow velocities, then owing to the Kelvin–Helmholtz (shear flow) instability, magnetohydrodynamic turbulence is generated (Chiueh & Zweibel 1987). This turbulence plays an important role in particle acceleration (Miller et al. 1996; Petrosian 2016). Some estimations of the level of this turbulence were made based on the broadening of the observed spectral lines (Hassler et al. 1990; Harra et al. 2013; Polito et al. 2018). In the paper by Karlický (2014) the frequency variations of the zebra stripes were interpreted as being caused by the plasma turbulence. These frequency variations were analyzed by the Fourier method and the spectra with the powerlaw index close to the Kolmogorov index were found.
In the present paper we estimate the levels of the time and spatially varying turbulence in the density and magnetic field in the zebra radio sources observed during solar flares. In particular, the estimation of the levels of the magnetic field turbulence is new.
We use the zebra model based on the doubleplasma resonance (DPR) instability (Zheleznyakov & Zlotnik 1975; Winglee & Dulk 1986; Kuznetsov & Tsap 2007; Tan 2010; Zlotnik 2013; Tan et al. 2014; Karlický & Yasnov 2015; Benáček & Karlický 2019). In this model, each zebrastripe source is located at a different place. It enables us to estimate the turbulence level not only in each zebrastripe source (time varying turbulence), but also the turbulence level in space between the zebrastripe sources at a given time (spatially varying turbulence).
The spatially varying turbulence between zebrastripe sources is a new aspect in the zebrastripe analysis. Moreover, it removes some problems with the DPR zebra model recently discussed in the paper by Yasnov & Chernov (2020). The authors calculated a sequence of the gyroharmonic numbers of zebra stripes in the whole zebra from the zebrastripe frequencies at the high and lowfrequency parts of the zebras. They found that the sequence of the gyroharmonic numbers derived from the high and lowfrequency zebra stripes differ in some zebras. Based on this, they concluded that these zebras cannot be explained by the DPR model. However, when we include the turbulence in the DPR model then the difference in calculated sequences of the gyroharmonic numbers can be explained within the DPR model. In the DPR model the zebrastripe sources are located at different positions, and thus the plasma turbulence modifies the zebrastripe frequencies of different zebra stripes in a different way. This introduces a difference in the sequence of the gyroharmonic numbers calculated from the high and lowfrequency parts of the zebras.
In this paper we also present a new method for determining the gyroharmonic numbers of the zebra stripes. The results obtained by this new method are verified by our previous method, presented in the paper by Karlický & Yasnov (2015). We apply these methods for the 14 February 1999 zebra event and calculate not only the gyroharmonic numbers of zebra stripes, but also the density and magnetic field. Then, knowing the gyroharmonic numbers of zebra stripes in this zebra event, we estimate the level of the time varying and spatially varying turbulence in the density and magnetic field in the zebrastripe sources. This enables us to compare these turbulence levels for the first time.
2. Method and its application
In our method we use the zebra model based on the doubleplasma resonance (DPR) instability (Zheleznyakov & Zlotnik 1975; Winglee & Dulk 1986; Zlotnik 2013; Benáček & Karlický 2018). Our method consists of two steps. First, we determine the gyroharmonic numbers of zebra stripes by a new method and by our previous methods (Karlický & Yasnov 2015). Second, knowing the gyroharmonic numbers and the zebrastripe frequencies, we calculate the density and magnetic field in the zebrastripe sources, and also their temporal and spatial variations, and thus the levels of their turbulence. The new method for determining the gyroharmonic numbers is highlighted by the example below where we use artificially generated zebrastripe frequencies. Then this new method is applied to the observed zebra.
In the DPR model the ratio of the two different zebrastripe frequencies can be expressed as (Karlický & Yasnov 2019)
where f_{s} and f_{s + m} are the zebrastripe frequencies for the zebra stripe with the gyroharmonic numbers s and s + m; R = L_{b}/L_{n}, where L_{n} and L_{b} is the density and magnetic field scale in the density n and magnetic field B profiles
assumed in the zebra source; n_{0} and B_{0} are the density and magnetic field at some reference level; and h means the distance along the axis of the source.
In order to highlight our new method for determining the gyroharmonic numbers of zebra stripes, we employ model (artificial) zebrastripe frequencies. We use relation (1) with R = 0.65, which corresponds to sequence A (Karlický & Yasnov 2018); i.e., the zebrastripe frequency in the zebra increases with decreasing gyroharmonic number. We take this sequence A because it agrees with all zebras analyzed so far. We define the gyroharmonic number s_{L} for the lowest zebrastripe frequency f_{L}. In our example, we take s_{L} = 30 for f_{L} = 1250 MHz. In the relation (1) we change m from −1 to −6, and calculate seven model zebrastripe frequencies (F_{model} = 1250, 1270, 1292, 1315, 1339, 1365, and 1392 MHz).
Solving Eq. (1) for R we express R as R_{1, 2} (i.e., in terms of the ratio of neighboring zebrastripe frequencies and corresponding gyroharmonic numbers). Thus, we obtain the relation as
Here R_{1, 2} is computed between the zebrastripe frequency 1 and higher neighboring zebrastripe frequency 2, starting from the lowest frequency f_{L}. In the following we make a number of computations of R_{1, 2} for all pairs of neighboring zebrastripe frequencies using the s_{L} as free parameter. We calculated R_{1, 2} for the F_{model} frequencies and for six values of s_{L} = 15, 20, 25, 30, 35, and 40.
In Fig. 1 we present the plots of these R_{1, 2} values in dependance on the gyroharmonic number s, including the regression lines. The red asterisks with the horizontal regression line in Fig. 1 correspond to the plot with s_{L} = 30. As expected in this case, in all computed pairs of frequencies R_{1, 2} is equal to 0.65, as in the case when the model zebrastripe frequencies were calculated from the relation (1) with s_{L} = 30 and R = 0.65. But, when we use a value of s_{L} lower than 30 to calculate R_{1, 2}, then the value of R_{1, 2} increases with increasing s. On the other hand, when we use a value of s_{L} greater than 30 to calculate R_{1, 2}, then the value of R_{1, 2} decreases with increasing s (see Fig. 1). It should be noted that we fitted all the plots of R_{1, 2} by the straight line – regression line. Except the case with s_{L} = 30, where R_{1, 2} lie on the regression line, the other plots of R_{1, 2} subtly deviate from the regression lines and the deviations increase with increasing s_{L} − 30. In the following, we utilize the change in the slope of the R_{1, 2} plots shown in Fig. 1 to calculate s_{L} and R in the observed zebra. The actual value of s_{L} is determined when the regression line in the R_{1, 2} data, calculated from the observed zebrastripe frequencies, has a minimum slope compared to the horizontal line. This solution also gives the value of R = constant.
Fig. 1. Ratio of R_{1, 2} in dependance on the gyroharmonic number s for six values of s_{L} = 15 (green asterisks), 20 (light blue), 25 (yellow), 30 (red), 35 (dark blue), and 40 (orange). Dashed lines mean the regression lines for all R_{1, 2}. 
For a calculation based on observations, we use the zebra recorded on 14 February 1999 by the Ondřejov radiospectrograph (Jiřička et al. 1993) (Fig. 2). As seen in this figure, the most distinct and continuous zebra stripes appear at approximately 12:08:57.0 UT. The zebrastripe frequencies at this time are 1548, 1573, 1599, 1619, 1642, 1674, 1707, 1739, 1770, 1810, 1845, and 1880 MHz. We use these frequencies to determine the gyrofrequency numbers. We vary s_{L} in a reasonable interval from 10 to 70, and calculate plots of R_{1, 2}. For each R_{1, 2} plot we compute the regression line in the same way as the variation in the slope of the R_{1, 2} plots in Fig. 1, looking for such s_{L} that the regression line has a minimum slope (i.e., this regression line represents R = constant). In Fig. 3b we show R_{1, 2} for the 14 February 1999 zebra versus s in the case when the regression line has the minimum slope. In this case s_{L} = 32 and the R corresponding to the regression (dashed) line is 0.625. For comparison, Figs. 3a and c show cases for s_{L} = 22 and s_{L} = 42; we see in this case that the slope of the regression lines change with s_{L}, exactly like those in Fig. 1.
Fig. 2. Radio spectrum showing the zebra observed at 14 February 1999. 
Fig. 3. R_{1, 2} in dependance on s, derived from the 14 February 1999 zebra: (a) for s_{L} = 22, (b) for s_{L} = 32, and (c) for s_{L} = 42. Dashed lines mean the regression lines. In case (b) the regression line has a minimum slope compared to the horizontal line. Thus, this case gives the resulting s_{L} = 32 and R = 0.625. 
We checked the resulting s_{L} and R using our previous method, which is described in Karlický & Yasnov (2015). The results from this previous method are the same (s_{L} = 32 and R = 0.625).
Now, knowing the gyroharmonic numbers of the zebra stripes and using the relations
where f_{s} is in MHz, we calculated the density n and magnetic field B in the zebrastripe sources at 12:08:57.0 UT. The results for each zebrastripe source, designated by s, are in Table 1.
Density and magnetic field in the zebrastripe sources at 12:08:57.0 UT in the 14 February 1999 zebra.
Now, let us estimate the level of turbulence in the zebra source. First, we estimate the level of the time varying turbulence. For this purpose we take the zebrastripe frequencies at eight zebra stripes in the eight equal consecutive windows covering the interval 12:08:56.7–12:08:57.4 UT, i.e., at a time around 12:08:57.0 UT where the gyroharmonic numbers were determined and where there are continuous zebra stripes. Using the relations in Eq. (4) we calculated the variations in density and magnetic field in the zebrastripe sources in comparison to their mean values. The results are shown in Fig. 4. The level of the time varying turbulence in density and magnetic field, expressed as the standard deviation for each window, is Δn/n_{t} = 0.0112–0.0149 and ΔB/B_{t} = 0.0056–0.0074.
Fig. 4. Time variations of the density Δn/n and magnetic field ΔB/B in the time interval 12:08:56.7–12:08:57.4 UT for the 14 February 1999 zebra. Black line: s = 28; violet: s = 27; green: s = 26; light blue: s = 25; yellow: s = 24; red: s = 23; blue: s = 22; and orange: s = 21. The vertical dashed line shows the time 12:08:57 UT, when the calculation of the gyroharmonic numbers was made. 
Now, let us estimate the level of the spatially varying turbulence. Figure 3b shows that R_{1, 2} varies, which means that the observed zebrastripe frequencies differ from those in the model, where R = 0.625 (dashed line). Although the error in zebrastripe frequency measurements, which is ∼2 MHz, can contribute to this variation, we assume that the variation is caused by the plasma turbulence in the zebrastripe sources.
Using the lowest zebrastripe frequency in the 14 February 1999 zebra (f_{L} = 1548 MHz) and computed values of s_{L} = 32 and R = 0.625 we generate the model frequencies according to relation (1). Thus, for all zebra stripes with s we know the observed and model frequencies. Using these frequencies and the relations in Eq. (4), we compute the frequency difference between the observed and modeled frequencies Δf and the variation in the density Δn/n and magnetic field ΔB/B in all zebrastripe sources. The results are shown in Fig. 5. The corresponding levels of the spatially varying turbulence expressed as the standard deviations are Δn/n_{s} = 0.0047 and ΔB/B_{s} = 0.0024.
Fig. 5. Frequency difference between the observed and modeled zebrastripe frequencies Δf, and variations in the density Δn/n and magnetic field ΔB/B in dependence on the gyroharmonic number s at 12:08:57 UT for the 14 February 1999 zebra. 
3. Discussion and conclusions
In this paper we presented not only a new method for estimating the plasma turbulence level in the zebrastripe radio sources, but also a new method for determining the gyroharmonic numbers s in zebras. The method used to determine s is equivalent to the method presented in Karlický & Yasnov (2015).
We estimated levels of the time and spatially varying turbulence in the zebra stripe sources. The estimated levels are Δn/n_{t} = 0.0112–0.0149 and ΔB/B_{t} = 0.0056–0.0074 for the time varying turbulence and Δn/n_{s} = 0.0047 and ΔB/B_{s} = 0.0024 for the spatially varying turbulence. These turbulence levels differ in that the level of the spatially varying turbulence is lower. It indicates some anisotropy of this turbulence, probably caused by the magnetic field structure around zebrastripe sources.
As seen in Figs. 4 and 5, variations in the density and magnetic field are in phase, which indicates the fast magnetosonic wave turbulence. The estimated turbulence levels are partly corrupted by the errors on the observed zebrastripe frequencies. The estimated error on this determination is about 2 MHz, to be compared with the maximum difference between the zebrastripe frequency and the mean zebrastripe frequency in the studied interval (12:08:56.7–12:08:57.4 UT) (∼20 MHz), and the maximum difference between model and observed zebrastripe frequencies (∼10 MHz; Fig. 5).
Figure 6 shows the regions of the plasma beta parameter greater or smaller than 1 in dependance on the plasma temperature and the gyroharmonic number s of zebra stripes. This plot is made using the plasma beta definition and the relations in Eq. (4). As seen in this figure, the plasma beta parameter in the zebrastripe sources is lower than 1 if the temperature is at least 1 MK (coronal temperature) and the gyroharmonic numbers is smaller than about 50. For higher gyroharmonic numbers and β ≤ 1, the temperature in the zebrastripe sources is lower than 1 MK. In our case of the 14 February 1999 zebra, the gyroharmonic number s_{L} is 32; therefore, the temperature in the zebra source for the plasma beta parameter β < 1 can be up to 3 MK.
Fig. 6. Regions of the plasma beta parameter β > 1 and β < 1 in the diagram of the plasma temperature vs. the gyroharmonic number s. 
Acknowledgments
The authors thank the referee for constructive comments that improved the article. We acknowledge support from the project RVO67985815 and GAČR grants 1809072S, 1909489S, 2009922J, and 2007908S. L. V. Yasnov acknowledges support from the Russian Foundation for Basic Research, Grants 182921016mk, and partly from Grant 180200045, from Program RAN No.28, Project 1D and State Task No. AAAAA171170118100134.
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All Tables
Density and magnetic field in the zebrastripe sources at 12:08:57.0 UT in the 14 February 1999 zebra.
All Figures
Fig. 1. Ratio of R_{1, 2} in dependance on the gyroharmonic number s for six values of s_{L} = 15 (green asterisks), 20 (light blue), 25 (yellow), 30 (red), 35 (dark blue), and 40 (orange). Dashed lines mean the regression lines for all R_{1, 2}. 

In the text 
Fig. 2. Radio spectrum showing the zebra observed at 14 February 1999. 

In the text 
Fig. 3. R_{1, 2} in dependance on s, derived from the 14 February 1999 zebra: (a) for s_{L} = 22, (b) for s_{L} = 32, and (c) for s_{L} = 42. Dashed lines mean the regression lines. In case (b) the regression line has a minimum slope compared to the horizontal line. Thus, this case gives the resulting s_{L} = 32 and R = 0.625. 

In the text 
Fig. 4. Time variations of the density Δn/n and magnetic field ΔB/B in the time interval 12:08:56.7–12:08:57.4 UT for the 14 February 1999 zebra. Black line: s = 28; violet: s = 27; green: s = 26; light blue: s = 25; yellow: s = 24; red: s = 23; blue: s = 22; and orange: s = 21. The vertical dashed line shows the time 12:08:57 UT, when the calculation of the gyroharmonic numbers was made. 

In the text 
Fig. 5. Frequency difference between the observed and modeled zebrastripe frequencies Δf, and variations in the density Δn/n and magnetic field ΔB/B in dependence on the gyroharmonic number s at 12:08:57 UT for the 14 February 1999 zebra. 

In the text 
Fig. 6. Regions of the plasma beta parameter β > 1 and β < 1 in the diagram of the plasma temperature vs. the gyroharmonic number s. 

In the text 
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