**Analysis of Long-Periodic Fluctuations of Solar Microwave Radiation, as a Way for Diagnostics of Coronal Magnetic Loops Dynamics**

Maxim L. Khodachenko1, Albert G. Kislyakov2 and Eugeny I. Shkelev2 *1Space Research Institute, Austrian Academy of Sciences, Graz, 2Lobachevsky State University, Nizhny Novgorod, 1Austria 2Russia* 

#### **1. Introduction**

142 Fourier Transform Applications

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The solar corona has a very complex and highly dynamic structure. It consists of a large number of constantly evolving, loops and filaments, which interact with each other and are closely associated with the local magnetic field. The non-stationary character of solar plasma-magnetic structures manifests itself in various forms of the coronal magnetic loops dynamics as rising motions, oscillations, meandering, twisting (Aschwanden et al., 1999; Schrijver et al., 1999), as well as in formation, sudden activation and eruption of filaments and prominences. Energetic phenomena, related to these types of magnetic activity, range from tiny transient brightenings (micro-flares) and jets to large, active-region-sized flares and coronal mass ejections (CMEs). They are naturally accompanied by different kinds of electromagnetic (EM) emission, covering a wide frequency band from radio waves to gamma-rays. Radiation, produced within a given plasma environment, carries an information on physical and dynamic conditions in a radiating source. This causes an exceptional importance of the EM radiation, as a diagnostic tool, for understanding the nature and physics of various solar dynamic phenomena. As a relatively new, in that context, direction of study in the traditional branch of the solar microwave radio astronomy appears the analysis of the slow, long-periodic (e.g., > 1 s) fluctuations of the radiation intensity (Khodachenko et al., 2005; Zaitsev et al., 2003).

Microwave radiation from the magnetic loops in solar active regions (e.g., during solar flares) is usually interpreted as a gyro-synchrotron radiation, produced by fast electrons on harmonics of the gyro-frequency *ν<sup>B</sup>* in the magnetic field *B* of the loop. In the case of a power-law distribution of electrons in energy as *<sup>f</sup>*(E) <sup>∝</sup> <sup>E</sup>−*δ*, the intensity of gyro-synchrotron radiation *I<sup>ν</sup>* from an optically thin loop (Dulk, 1985; Dulk & Marsh, 1982) is

$$I\_{\nu} \propto B^{-0.22 + 0.9\delta} (\sin \theta)^{-0.43 + 0.65\delta} \,\prime \tag{1}$$

where *θ* is the angle between magnetic field and the direction of electromagnetic wave propagation. For the observed typical values of the electron energy spectrum index 2 ≤ *δ* ≤ 7 this implies the proportionality of intensity to a moderately high power of the background

ambient plasma and details of the magnetic structure. The LCR model approach was applied in particular for interpretation of the solar microwave burst long-periodic modulations with drifting modulation frequencies *ν*(*t*) in the interval 0.03 ÷ 1 Hz. Based on the analysis of the frequency drift of the modulations Zaitsev et al. (1998; 2001a; 2003) and Khodachenko et al. (2005) estimated also the values of the electric current in flaring loops (10<sup>8</sup> <sup>÷</sup> 1011 A), which appeared to be close to the values obtained by other methods (Hardy et al., 1998; Leka et al.,

<sup>145</sup> Analysis of Long-Periodic Fluctuations of Solar

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

However, not all long-periodic modulations of solar microwave radiation demonstrate frequency drift and occupy a similar frequency range. Khodachenko et al. (2009) pointed out that the frequency of LCR-oscillations of the electric current, which depends on specific parameters of a coronal loop, usually stays within the interval *νLCR* ≈ (0.03 ÷ 1) Hz. Therefore, the modulations of the solar microwave radiation intensity with 0.03Hz< *ν* < 1Hz are very likely to be due to the electric currents, oscillating in LCR-circuits of coronal loops. At the same time, modulations caused by the oscillatory motions of loops that contain the radiation sources, because of their direct connection with the large-scale dynamics of loops, should have typical frequencies < 0.01 Hz and exhibit no drift. Thus, it has been proposed in (Khodachenko et al., 2009) that one should distinguish, when speaking about different kinds of long-periodic modulations of the solar microwave radiation, between the *low-frequency (LF)* (≈ 0.03 ÷ 1 Hz) and *very-low-frequency (VLF)* (< 0.01 Hz) modulations, assuming the first to be connected with the LCR-oscillations of electric currents in the coronal loops and the second to be caused by large-scale motions of the radiation sources confined within the oscillating loops. LF modulations (e.g., 0.03 ÷ 1 Hz) have been studied in details and interpreted in terms of the equivalent electric circuit models of coronal loops in Khodachenko et al. (2005; 2006); Zaitsev et al. (1998; 2001a;b; 2003). They are not considered in this chapter, whereas we addresses here the VLF modulations (< 0.01 Hz) of solar microwave radiation and their possible relation to the large-scale dynamics of coronal loops in solar active regions. Some preliminary results on that subject were reported recently in a short publication Khodachenko et al. (2011), and the

Transverse oscillations of the coronal magnetic loops considered here are triggered by flares and filament eruptions, i.e. by phenomena in which the significant Lorentz forces are likely acting in association with the magnetic field adjustments. By this, all considered oscillating loops have at least one footpoint in the immediate vicinity of a separatrix surface or of a flare ribbon. Using the potential-field extrapolations Schrijver & Brown (2000) demonstrated that the field lines close to a separatrix surface exhibit strongly amplified displacements in response to small displacements in the photospheric "roots". This means that the magnetic field lines in the proximity of separatrix are much more sensitive to changes in the field sources than are the field lines that lie well within domains of connectivity. Speaking about a nature of the observed transversal oscillations of coronal loops Schrijver et al. (2002) address two models: (a) transverse waves in coronal loops that act as wave guides and (b) mentioned above, strong sensitivity of the shape of magnetic field lines near separatrix to changes in the bottom field sources. By this, the authors outline several observational features that favor the model (b). Based on the extensive study of properties of transverse loop oscillations triggered by flares, Aschwanden et al. (2002) also concluded that most of the loops do not fit the simple model of a kink eigen-mode oscillation. Therefore, the present paper is not dedicated to the study of MHD oscillations in solar coronal loops. Our goal consists in demonstration of the fact that quasi-periodic transverse motions of a coronal magnetic loop, which contains

1996; Moreton & Severny, 1968; Spangler, 2007; Tan et al., 2006).

present paper addresses this topic in more details.

magnetic field and essential anisotropy of the radiation: *I<sup>ν</sup>* ∝ *B*1.58÷6.08( sin *θ*)0.87÷4.12. Equation (1) is obtained within an assumption of an optically thin source, when the radiation intensity is proportional to the emissivity *ην* (Dulk, 1985). According to the estimations in Urpo et al. (1994), a coronal loop with diameter of about 4 <sup>×</sup> 108 cm is optically thin in the considered frequency range for the gyro-synchrotron absorption if the density of fast electrons is <sup>&</sup>lt; <sup>2</sup> <sup>×</sup> <sup>10</sup><sup>9</sup> cm−3. The typical density of <sup>&</sup>gt; 10 keV electrons in the microwave burst events is usually 10<sup>6</sup> <sup>÷</sup> <sup>10</sup><sup>7</sup> cm−<sup>3</sup> and therefore, it stays well within the above indicated limit.

It follows from the equation (1) that variations of the loop magnetic field, associated with disturbances of the electric current in a radiating source, should modulate the intensity of the microwave radiation of the loop (Khodachenko et al., 2005; Zaitsev et al., 2003). Another origin for the modulation of intensity of the observed microwave radiation can be due to the quasi-periodic motion (oscillation) of a coronal magnetic loop, containing the radiation source. This mechanism is connected with the anisotropy of the gyro-synchrotron emission, as well as with the variation of the magnetic field value during the oscillatory motion of the loop, which both, according to the equation (1), can result in a quasi-periodic modulation of the received signal (Khodachenko et al., 2006; 2011). Therefore, the analysis of slow modulations of solar microwave radiation may be used for the diagnostics of oscillating electric currents in the coronal loops, as well as for the investigation of large-scale motion of the loops (including loop oscillations) in solar active regions. By this, it is natural to expect that structural complexity of solar active regions will manifest itself in peculiarities of the emitted radiation.

The dynamic spectra of the long-periodic oscillations, modulating the intensity of microwave radiation from solar active regions, have been found to contain quite often several spectral tracks, demonstrating a specific temporal behaviour (Zaitsev et al., 1998; 2001a;b; 2003). Khodachenko et al. (2005) considered these multi-track features as an indication that the detected microwave radiation is produced within a system of a few closely located, magnetic loops, having slightly different parameters and involved in a kind of common global dynamic process. In several cases such slow modulations of solar microwave radiation (with multi-track spectra) were interpreted as the signatures of oscillating electric currents, running within the circuits of moving relative each other inductively connected coronal magnetic loops (Khodachenko et al., 2006). The dynamics of these electric currents has been described by means of the equivalent electric circuit (LCR-circuit) models of the coronal loops (Khodachenko et al., 2003; Zaitsev et al., 1998) characterized by time-dependent inductance *L*, capacitance *C*, resistance *R*, as well as mutual inductance coefficients *Mj* (Khodachenko et al., 2003; 2009). The *L*, *C*, *R* and *Mj* parameters of electric circuit of a current-carrying loop depend on shape, scale, position of the loop with respect to other loops, as well as on the plasma parameters and value of the total longitudinal current in the magnetic tube. In that respect it is worth to mention that the LCR-circuit model ignores the fact that changes of the magnetic field and related electric current propagate in plasma at the Alfvén speed. It ignores any short-time variations of plasma parameters, which appear to be averaged in course of derivation of the LCR model equation (Khodachenko et al., 2009; Zaitsev et al., 2001a). The LCR approach assumes instant changes of the electric current over the whole electric circuit according to the varying potential and ignores all the "propagation effects" related to the system MHD modes. The LCR equations correctly describe temporal evolution of electric currents in a system of solar magnetic current-carrying loops only at a time scale longer than the Alfvén wave propagation time. More generally, the equivalent electric circuit model of a coronal loop tends to emphasize the global electric circuit, obscuring the effects of the 2 Will-be-set-by-IN-TECH

magnetic field and essential anisotropy of the radiation: *I<sup>ν</sup>* ∝ *B*1.58÷6.08( sin *θ*)0.87÷4.12. Equation (1) is obtained within an assumption of an optically thin source, when the radiation intensity is proportional to the emissivity *ην* (Dulk, 1985). According to the estimations in Urpo et al. (1994), a coronal loop with diameter of about 4 <sup>×</sup> 108 cm is optically thin in the considered frequency range for the gyro-synchrotron absorption if the density of fast electrons is <sup>&</sup>lt; <sup>2</sup> <sup>×</sup> <sup>10</sup><sup>9</sup> cm−3. The typical density of <sup>&</sup>gt; 10 keV electrons in the microwave burst events is

It follows from the equation (1) that variations of the loop magnetic field, associated with disturbances of the electric current in a radiating source, should modulate the intensity of the microwave radiation of the loop (Khodachenko et al., 2005; Zaitsev et al., 2003). Another origin for the modulation of intensity of the observed microwave radiation can be due to the quasi-periodic motion (oscillation) of a coronal magnetic loop, containing the radiation source. This mechanism is connected with the anisotropy of the gyro-synchrotron emission, as well as with the variation of the magnetic field value during the oscillatory motion of the loop, which both, according to the equation (1), can result in a quasi-periodic modulation of the received signal (Khodachenko et al., 2006; 2011). Therefore, the analysis of slow modulations of solar microwave radiation may be used for the diagnostics of oscillating electric currents in the coronal loops, as well as for the investigation of large-scale motion of the loops (including loop oscillations) in solar active regions. By this, it is natural to expect that structural complexity of

The dynamic spectra of the long-periodic oscillations, modulating the intensity of microwave radiation from solar active regions, have been found to contain quite often several spectral tracks, demonstrating a specific temporal behaviour (Zaitsev et al., 1998; 2001a;b; 2003). Khodachenko et al. (2005) considered these multi-track features as an indication that the detected microwave radiation is produced within a system of a few closely located, magnetic loops, having slightly different parameters and involved in a kind of common global dynamic process. In several cases such slow modulations of solar microwave radiation (with multi-track spectra) were interpreted as the signatures of oscillating electric currents, running within the circuits of moving relative each other inductively connected coronal magnetic loops (Khodachenko et al., 2006). The dynamics of these electric currents has been described by means of the equivalent electric circuit (LCR-circuit) models of the coronal loops (Khodachenko et al., 2003; Zaitsev et al., 1998) characterized by time-dependent inductance *L*, capacitance *C*, resistance *R*, as well as mutual inductance coefficients *Mj* (Khodachenko et al., 2003; 2009). The *L*, *C*, *R* and *Mj* parameters of electric circuit of a current-carrying loop depend on shape, scale, position of the loop with respect to other loops, as well as on the plasma parameters and value of the total longitudinal current in the magnetic tube. In that respect it is worth to mention that the LCR-circuit model ignores the fact that changes of the magnetic field and related electric current propagate in plasma at the Alfvén speed. It ignores any short-time variations of plasma parameters, which appear to be averaged in course of derivation of the LCR model equation (Khodachenko et al., 2009; Zaitsev et al., 2001a). The LCR approach assumes instant changes of the electric current over the whole electric circuit according to the varying potential and ignores all the "propagation effects" related to the system MHD modes. The LCR equations correctly describe temporal evolution of electric currents in a system of solar magnetic current-carrying loops only at a time scale longer than the Alfvén wave propagation time. More generally, the equivalent electric circuit model of a coronal loop tends to emphasize the global electric circuit, obscuring the effects of the

usually 10<sup>6</sup> <sup>÷</sup> <sup>10</sup><sup>7</sup> cm−<sup>3</sup> and therefore, it stays well within the above indicated limit.

solar active regions will manifest itself in peculiarities of the emitted radiation.

ambient plasma and details of the magnetic structure. The LCR model approach was applied in particular for interpretation of the solar microwave burst long-periodic modulations with drifting modulation frequencies *ν*(*t*) in the interval 0.03 ÷ 1 Hz. Based on the analysis of the frequency drift of the modulations Zaitsev et al. (1998; 2001a; 2003) and Khodachenko et al. (2005) estimated also the values of the electric current in flaring loops (10<sup>8</sup> <sup>÷</sup> 1011 A), which appeared to be close to the values obtained by other methods (Hardy et al., 1998; Leka et al., 1996; Moreton & Severny, 1968; Spangler, 2007; Tan et al., 2006).

However, not all long-periodic modulations of solar microwave radiation demonstrate frequency drift and occupy a similar frequency range. Khodachenko et al. (2009) pointed out that the frequency of LCR-oscillations of the electric current, which depends on specific parameters of a coronal loop, usually stays within the interval *νLCR* ≈ (0.03 ÷ 1) Hz. Therefore, the modulations of the solar microwave radiation intensity with 0.03Hz< *ν* < 1Hz are very likely to be due to the electric currents, oscillating in LCR-circuits of coronal loops. At the same time, modulations caused by the oscillatory motions of loops that contain the radiation sources, because of their direct connection with the large-scale dynamics of loops, should have typical frequencies < 0.01 Hz and exhibit no drift. Thus, it has been proposed in (Khodachenko et al., 2009) that one should distinguish, when speaking about different kinds of long-periodic modulations of the solar microwave radiation, between the *low-frequency (LF)* (≈ 0.03 ÷ 1 Hz) and *very-low-frequency (VLF)* (< 0.01 Hz) modulations, assuming the first to be connected with the LCR-oscillations of electric currents in the coronal loops and the second to be caused by large-scale motions of the radiation sources confined within the oscillating loops. LF modulations (e.g., 0.03 ÷ 1 Hz) have been studied in details and interpreted in terms of the equivalent electric circuit models of coronal loops in Khodachenko et al. (2005; 2006); Zaitsev et al. (1998; 2001a;b; 2003). They are not considered in this chapter, whereas we addresses here the VLF modulations (< 0.01 Hz) of solar microwave radiation and their possible relation to the large-scale dynamics of coronal loops in solar active regions. Some preliminary results on that subject were reported recently in a short publication Khodachenko et al. (2011), and the present paper addresses this topic in more details.

Transverse oscillations of the coronal magnetic loops considered here are triggered by flares and filament eruptions, i.e. by phenomena in which the significant Lorentz forces are likely acting in association with the magnetic field adjustments. By this, all considered oscillating loops have at least one footpoint in the immediate vicinity of a separatrix surface or of a flare ribbon. Using the potential-field extrapolations Schrijver & Brown (2000) demonstrated that the field lines close to a separatrix surface exhibit strongly amplified displacements in response to small displacements in the photospheric "roots". This means that the magnetic field lines in the proximity of separatrix are much more sensitive to changes in the field sources than are the field lines that lie well within domains of connectivity. Speaking about a nature of the observed transversal oscillations of coronal loops Schrijver et al. (2002) address two models: (a) transverse waves in coronal loops that act as wave guides and (b) mentioned above, strong sensitivity of the shape of magnetic field lines near separatrix to changes in the bottom field sources. By this, the authors outline several observational features that favor the model (b). Based on the extensive study of properties of transverse loop oscillations triggered by flares, Aschwanden et al. (2002) also concluded that most of the loops do not fit the simple model of a kink eigen-mode oscillation. Therefore, the present paper is not dedicated to the study of MHD oscillations in solar coronal loops. Our goal consists in demonstration of the fact that quasi-periodic transverse motions of a coronal magnetic loop, which contains

be associated with the main and double frequency of the loop oscillation (*ν*<sup>0</sup> and 2*ν*0) may

<sup>147</sup> Analysis of Long-Periodic Fluctuations of Solar

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

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>

*δ* = 5 for different viewing angles Θ<sup>0</sup> = *π*/2; *π*/3; *π*/4; *π*/6 are shown in Figure 2.

stress factors work synchronously.

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

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

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

<sup>0</sup> sin2(2*πν*0*t*)).

indicate about a transverse oscillatory dynamics of the loop.

a source of microwave emission, may be connected with a specific modulation of radiation intensity received by a remote observer.

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.
