**2. Peculiarities of VLF modulations of microwave radiation related to large-scale transverse motions of the radiating sources in the oscillating coronal loops**

Even taking into account the typical ranges of the modulation frequencies of microwave radiation (mentioned above), it is usually difficult to identify the modulation mechanism acting in each particular case. Indeed, having in mind only Equation (1) one cannot say for sure if the observed modulation is due to *a)* the electric current oscillations in the radiation source or *b)* the large-scale oscillatory motion of the loop. However, regarding the last modulation mechanism, an attention should be paid to the fact that large-scale transverse oscillatory motion of a coronal loop is accompanied by the periodic stress of magnetic field created in the loop, especially near its footpoints, during each inclination, i.e. two times per oscillation cycle. This means that the magnetic field strength fluctuates during the oscillatory motion of the loop with a half-period *P*osc/2 of the loop oscillation. Therefore, according to Equation (1), for a transverse oscillating loop, a properly located observer, in addition to the modulation caused by the emission diagram motion at the main oscillation frequency *ν*<sup>0</sup> = 1/*P*osc, may see in some cases the modulation at the double frequency of the loop oscillation 2*ν*0, as well as weak higher order harmonics caused by the non-linearity of Equation (1). However, as it will be shown below the relative amplitude of the higher-order harmonics, i.e., with numbers > 2 is rather low, and in the most cases only the first two harmonic frequencies can be detected. The domination of the main and double-frequency harmonics in the spectrum is caused by implicit presence of these frequencies in the signal, according to the above described character of the radiation modulating factors. Therefore, the presence of the "modulation pairs" in the low-frequency spectra, i.e., the lines which can 4 Will-be-set-by-IN-TECH

a source of microwave emission, may be connected with a specific modulation of radiation

Speaking about other possible mechanisms (besides of the microwave radiation source large-scale oscillatory motion), which may cause a quasi-periodic modulation of the non-thermal electron gyro-synchrotron radiation, it is necessary to mention that a quasi-periodically varying flow of the non-thermal electrons may also result in oscillations of intensity of microwave radiation. Generation of energetic electrons usually is believed to be associated with the processes of magnetic reconnection during solar flares (Miller et al., 1997). There are also theories which suggest acceleration of particles by the inductive and charge separation electric fields, build in course of the continuous motion of solar large-scale coronal magnetic structures (Khodachenko et al., 2003; Zaitsev & Stepanov, 1992). Besides of that, particle acceleration in a collapsing magnetic trap (Karlický & Kosugi, 2004), in the MHD turbulence (LaRosa & Moore, 1993; Miller et al., 1996), and in shocks (Cargill et al., 1988; Holman & Pesses, 1983) are addressed as secondary possible mechanisms for energetic particle production. An extended review of particle acceleration processes in solar flares was recently published by Aschwanden (2002). In most of these cases the typical periods are shorter than those of the VLF transverse oscillations of coronal loops. On the other hand, there are also models in which VLF large-scale oscillations of coronal loops control the process of generation of energetic particles after the impulsive phase of a flare (Nakariakov et al., 2006). This case, however, deserves a special study, which appears beyond the scope of the present paper. Our analysis here is based on the traditional scenario, according to which the non-thermal particles, produced during a flare in particle acceleration regions (e.g., sites of magnetic reconnection or the area of separatrix currents), are injected into oscillating loops.

**2. Peculiarities of VLF modulations of microwave radiation related to large-scale transverse motions of the radiating sources in the oscillating coronal loops** Even taking into account the typical ranges of the modulation frequencies of microwave radiation (mentioned above), it is usually difficult to identify the modulation mechanism acting in each particular case. Indeed, having in mind only Equation (1) one cannot say for sure if the observed modulation is due to *a)* the electric current oscillations in the radiation source or *b)* the large-scale oscillatory motion of the loop. However, regarding the last modulation mechanism, an attention should be paid to the fact that large-scale transverse oscillatory motion of a coronal loop is accompanied by the periodic stress of magnetic field created in the loop, especially near its footpoints, during each inclination, i.e. two times per oscillation cycle. This means that the magnetic field strength fluctuates during the oscillatory motion of the loop with a half-period *P*osc/2 of the loop oscillation. Therefore, according to Equation (1), for a transverse oscillating loop, a properly located observer, in addition to the modulation caused by the emission diagram motion at the main oscillation frequency *ν*<sup>0</sup> = 1/*P*osc, may see in some cases the modulation at the double frequency of the loop oscillation 2*ν*0, as well as weak higher order harmonics caused by the non-linearity of Equation (1). However, as it will be shown below the relative amplitude of the higher-order harmonics, i.e., with numbers > 2 is rather low, and in the most cases only the first two harmonic frequencies can be detected. The domination of the main and double-frequency harmonics in the spectrum is caused by implicit presence of these frequencies in the signal, according to the above described character of the radiation modulating factors. Therefore, the presence of the "modulation pairs" in the low-frequency spectra, i.e., the lines which can

intensity received by a remote observer.

be associated with the main and double frequency of the loop oscillation (*ν*<sup>0</sup> and 2*ν*0) may indicate about a transverse oscillatory dynamics of the loop.

Formation of the "modulation pairs" and their higher-order harmonic companions in multi-line dynamic spectrum of the VLF modulation of microwave radiation emitted from a transverse oscillating coronal loop may be illustrated with a simple model. Let's suppose that the loop undergoes oscillations in the direction transverse to the loop plane as shown in Figure 1. The loop inclination relative to the vertical direction varies as *α*(*t*) = *α*<sup>0</sup> sin(2*πν*0*t*), where *α*<sup>0</sup> and *ν*<sup>0</sup> are the angular amplitude and frequency of the loop oscillations, respectively. Assuming that this loop, when oriented vertically, is seen by a remote observer at the angle Θ0, we get that in course of the loop oscillation the viewing angle changes as Θ(*t*) = Θ<sup>0</sup> − *α*(*t*).

Irrespectively of the nature of a coronal loop oscillation, the important feature of the large-scale transverse motion of the loop, consists in an oscillating magnetic stress, created in the loop during its quasi-periodic inclinations. Assuming the local transverse disturbance of the magnetic field relative its initial vertical direction to be *δB*, we find that the total disturbed magnetic field is *δB*/ cos *α*(*t*). For sufficiently small *α*(*t*) the following approximation can be used: 1/(cos *<sup>α</sup>*(*t*)) <sup>≈</sup> 1/(<sup>1</sup> <sup>−</sup> *<sup>α</sup>*(*t*)2)1/2 <sup>≈</sup> <sup>1</sup> + (1/2)*α*(*t*)2. This means that the disturbed magnetic field in the loop varies in time as *<sup>B</sup>*(*t*) <sup>≈</sup> *<sup>δ</sup>B*(<sup>1</sup> + (1/2)*α*(*t*)2). Therefore, for the assumed above sinusoidal character of *α*(*t*), we finally obtain that the local magnetic field in a transverse oscillating magnetic loop may be approximated as *B*(*t*) ∝ (1 + 0.5*α*<sup>2</sup> <sup>0</sup> sin2(2*πν*0*t*)). Substitution of the expressions for Θ(*t*) and *B*(*t*) into (1) enables to construct a modeling signal for the varying intensity of microwave radiation emitted from a transverse oscillating magnetic loop. The examples of dynamic spectra of this signal obtained with *α*<sup>0</sup> = *π*/6 and *δ* = 5 for different viewing angles Θ<sup>0</sup> = *π*/2; *π*/3; *π*/4; *π*/6 are shown in Figure 2.

Dynamical spectra of the modeling signal in Figure 2 demonstrate several important features, typical for the radiation emitted from a microwave source located in a transverse large-scale oscillating magnetic loop, which may be observed in the solar microwave emission. In particular, for the most of the viewing angles (except of Θ<sup>0</sup> = *π*/2) the dynamic spectra contain well pronounced "modulation pairs", e.g. the lines at the main *ν*<sup>0</sup> and double 2*ν*<sup>0</sup> frequency of the oscillation. Besides of that, sometimes also a weak third harmonic at 3*ν*<sup>0</sup> may be observed, which appears due to essentially non-sinusoidal (non-harmonic) character of the signal resulted from the joint action of two modulating factors: quasi-periodic magnetic stress and emission diagram motion. In a special case of Θ<sup>0</sup> = *π*/2, the absence of the main frequency component is caused by a "symmetrizing" (in this case) of the varying angular part of the emission intensity. This results in a situation when the diagram motion and magnetic stress factors work synchronously.

As it can be seen in Figure 2, only the first two harmonics have high enough amplitudes. In particular, the spectral amplitude of third harmonic in the cases with Θ<sup>0</sup> = *π*/3; *π*/4; *π*/6, never exceeds 25% of the main frequency component, whereas the second harmonic constitutes usually about 65% of the last. Therefore, the detection of harmonics with numbers higher than 2 in a natural signal, will be in the most cases difficult due to the noise contamination. The presence of modulation pairs in the VLF spectra of solar microwave radiation may be considered as an imprint of a transverse kink-type motion of a loop containing the radiation source. This feature may be used for the indirect identification of candidates for transverse oscillating coronal loops by finding specific modulation lines in the VLF dynamic spectra of microwave radiation. However, the exact detection of transverse

Fig. 2. Dynamical spectra of the modeling signal with *α*<sup>0</sup> = *π*/6 and *δ* = 5 demonstrating the peculiarities of the microwave radiation VLF modulations produced due to large-scale transverse oscillations of a coronal loop containing the radiating source. Different directions to an observer Θ<sup>0</sup> (viewing angles) are considered: (a) *π*/2; (b) *π*/3; (c) *π*/4; (d) *π*/6.

<sup>149</sup> Analysis of Long-Periodic Fluctuations of Solar

Microwave Radiation, as a Way for Diagnostics of Coronal Magnetic Loops Dynamics

Differently to the idealized infinite in time analytical modelling signal considered in section 2, the natural radio emission received from a solar active region with oscillating coronal loop(s) is essentially time-dependent. It usually begins with an impulsive phase of a solar flare and has duration of only several periods of decaying oscillations of the loop(s). Besides of that, a signal from a particular oscillating loop is quite often strongly contaminated by interfering signals from neighboring loops in the active region of interest, as well as by the radiations emitted from other solar active regions. This complicates the task of detection and diagnostics of coronal magnetic loop oscillations in microwaves and requires special data preparation procedures with consequent application of high spectral and time resolution data analysis

Since the analyzed data appear in a form of discrete counts of the signal intensity, it is natural that digital methods are applied for their processing. A basic specifics of the digital methods consists in certain limitation of dynamical range of the resulting spectra which may lead to the loss of relatively weak and short-time, but important parts of the whole spectra-temporal picture of the studied phenomenon. To avoid of that, the analyzed data pass certain pre-processing preparation which (depending on particular case and task) may include the following procedures: 1) Subtraction of a constant component of a signal, or the signal average; 2) subtraction of a slow (as compared to the analyzed oscillations) major trend of the signal; 3) slow polynomial approximation of the analyzed data with the consequent subtraction of the approximating signal; 4) signal "normalization" (will be described below);

**3. Data preparation and analysis methods**

techniques.

Fig. 1. Schematic view of an oscillating loop which contains a microwave radiation source. Θ<sup>0</sup> is the direction to a remote observer and *α*<sup>0</sup> is the angular amplitude of the loop oscillations.

motion of the radiating loops by the dynamical spectra of microwave emission (with the exclusion of other mechanisms which may also generate higher spectral harmonics) requires quantitative study of the measured radio signal and superimposing these results with the precise calculation of the radiation from the loop, taking into account the loop position relative observer.

Several real observational examples are considered below, for which VLF modulations of microwave radiation could be associated with the observed in EUV post-flare oscillating coronal loops.

6 Will-be-set-by-IN-TECH

Fig. 1. Schematic view of an oscillating loop which contains a microwave radiation source. Θ<sup>0</sup> is the direction to a remote observer and *α*<sup>0</sup> is the angular amplitude of the loop

motion of the radiating loops by the dynamical spectra of microwave emission (with the exclusion of other mechanisms which may also generate higher spectral harmonics) requires quantitative study of the measured radio signal and superimposing these results with the precise calculation of the radiation from the loop, taking into account the loop position relative

Several real observational examples are considered below, for which VLF modulations of microwave radiation could be associated with the observed in EUV post-flare oscillating

oscillations.

observer.

coronal loops.

Fig. 2. Dynamical spectra of the modeling signal with *α*<sup>0</sup> = *π*/6 and *δ* = 5 demonstrating the peculiarities of the microwave radiation VLF modulations produced due to large-scale transverse oscillations of a coronal loop containing the radiating source. Different directions to an observer Θ<sup>0</sup> (viewing angles) are considered: (a) *π*/2; (b) *π*/3; (c) *π*/4; (d) *π*/6.
