# Simulations of Gyrosynchrotron Microwave Emission from an Oscillating 3D
Magnetic Loop

A.A. Kuznetsov1T. Van Doorsselaere2V.E. Reznikova2,3 1 Institute of Solar-
Terrestrial Physics, Irkutsk 664033, Russia e-mail<EMAIL_ADDRESS>
2 CmPA, Department of Mathematics, KU Leuven, BE-3001 Leuven, Belgium e-mail:
<EMAIL_ADDRESS>
3 Radiophysical Research Institute (NIRFI), Nizhny Novgorod 603950, Russia

###### Abstract

Radio observations of solar flares often reveal various periodic or quasi-
periodic oscillations. Most likely, these oscillations are caused by
magnetohydrodynamic (MHD) oscillations of flaring loops which modulate the
emission. Interpretation of the observations requires comparing them with
simulations. We simulate the gyrosynchrotron radio emission from a semi-
circular (toroidal-shaped) magnetic loop containing sausage-mode MHD
oscillations. The aim is to detect the observable signatures specific to the
considered MHD mode and to study their dependence on the various source
parameters. The MHD waves are simulated using a linear three-dimensional model
of a magnetized plasma cylinder; both standing and propagating waves are
considered. The curved loop is formed by replicating the MHD solutions along
the plasma cylinder and bending the cylinder; this model allows us to study
the effect of varying the viewing angle along the loop. The radio emission is
simulated using a three-dimensional model and its spatial and temporal
variations are analyzed. We consider several loop orientations and different
parameters of the magnetic field, plasma, and energetic electrons in the loop.
In the model with low plasma density, the intensity oscillations at all
frequencies are synchronous (with the exception of a narrow spectral region
below the spectral peak). In the model with high plasma density, the emission
at low frequencies (where the Razin effect is important) oscillates in anti-
phase with the emissions at higher frequencies. The oscillations at high and
low frequencies are more pronounced in different parts of the loop (depending
on the loop orientation). The layers where the line-of-sight component of the
magnetic field changes sign can produce additional peculiarities in the
oscillation patterns.

###### keywords:

Coronal seismology — Flares, waves — Oscillations, solar — Radio bursts,
microwave — Waves, magnetohydrodynamic

##  1 Introduction

Waves and oscillations are a common phenomeno in solar flares. They are
believed to reveal themselves as quasi-periodic pulsations in spatially
unresolved observations at different wavelengths (see, e.g., the review of
Nakariakov and Melnikov, 2009) and have been directly detected using the EUV
observations with high spatial resolution (Nakariakov et al., 1999; Aschwanden
et al., 1999; De Moortel and Nakariakov, 2012). In particular, the observed
periodicities of the quasi-periodic pulsations in the radio emission, hard
X-rays, and $\gamma$-rays (from a fraction of a second to several minutes)
agree with the expected magnetohydrodynamic (MHD) timescales which strongly
suggests a close relationship between the pulsations and the MHD waves and
oscillations (Nakariakov and Melnikov, 2009; De Moortel and Nakariakov, 2012).
It should be noted, however, that the solar quasi-periodic pulsations are
usually superimposed on a rapidly varying background, so that recovering their
parameters is not easy; some previously reported conclusions may be unreliable
due to the methodology used (Gruber et al., 2011).

Observations of the waves and oscillations in the flares can be used to
diagnose the quasi-periodic magnetic reconnection processes that are believed
to be the most probable source of these oscillations (e.g., Kliem, Karlický,
and Benz, 2000). The MHD waves and oscillations can also be used to diagnose
the coronal plasma and magnetic field parameters (“coronal seismology”; see,
e.g., Nakariakov and Verwichte, 2005). Moreover, the waves and oscillations
are believed to play a key role in the energy transfer both in the quiet
corona (e.g., Erdélyi, 2006) and in the flares (Fletcher and Hudson, 2008);
they can modulate and/or trigger the magnetic reconnection processes (Chen and
Priest, 2006; Doyle, Popescu, and Taroyan, 2006; Nakariakov et al., 2006) and
thus affect greatly the behaviour of unstable active regions and development
of the solar flares.

One of the potential observable manifestations of MHD waves is quasi-periodic
pulsations of radio emission (e.g., Aschwanden, 1987; Nakariakov and Melnikov,
2009; Kupriyanova et al., 2010), because these waves can modulate both the
coherent and incoherent radio emission mechanisms. Interpretation of the
observed quasi-periodic pulsations requires comparing the observations with
simulations in order to determine the origin of the pulsations (because they
can be caused not only by the MHD waves but also, e.g., by quasi-periodic
injection of energetic particles); if the oscillations are due to waves, it
allows us to identify the wave mode and other characteristics of the MHD
waves. In turn, identification of the wave mode is of high importance for the
development of the coronal seismology. As a rule, the basic characteristics of
the pulsations (such as the period) have no unique interpretation and
therefore some more subtle features (such as the relations between the
pulsations observed in different spectral channels) have to be analyzed
(Fleishman, Bastian, and Gary, 2008; Mossessian and Fleishman, 2012).

Perhaps the best tool to study the physical parameters and processes in solar
flares is their microwave emission, because it is produced mainly due to the
incoherent gyrosynchrotron radiation mechanism which is (a) relatively simple
and well studied (Melrose, 1968; Ramaty, 1969) and (b) highly sensitive to the
magnetic field and plasma parameters. There are a lot of papers dedicated to
simulating the gyrosynchrotron emission in solar flares. In particular, two-
dimensional (2D) models (e.g., Alissandrakis and Preka-Papadema, 1984; Klein
and Trottet, 1984; Preka-Papadema and Alissandrakis, 1988; Kucera et al.,
1993; Kundu et al., 2001) and three-dimensional (3D) models (e.g., Preka-
Papadema and Alissandrakis, 1992; Bastian, Benz, and Gary, 1998; Nindos et
al., 2000; Simões and Costa, 2006; Costa et al., 2013) have been employed to
account for the structure of the emitting region. Some simulations involve the
magnetic field structure obtained by extrapolating the observed photospheric
magnetograms (e.g., Gary, Fleishman, and Nita, 2013; Kuznetsov and Kontar,
2015), take into account anisotropy of the emitting electrons (e.g., Fleishman
and Melnikov, 2003; Altyntsev et al., 2008; Simões and Costa, 2010; Kuznetsov,
Nita, and Fleishman, 2011) and/or evolution of the electron distributions
during flares (e.g., Tzatzakis, Nindos, and Alissandrakis, 2008; Reznikova et
al., 2009; Kuznetsov and Zharkova, 2010; Kuznetsov and Melnikov, 2012).
However, possible modulation of the emission by MHD waves has been studied
only in a few works and by using relatively simple models (Kopylova, Stepanov,
and Tsap, 2002; Nakariakov and Melnikov, 2006; Reznikova et al., 2007;
Mossessian and Fleishman, 2012). Recently, Reznikova, Antolin, and Van
Doorsselaere (2014) have simulated the gyrosynchrotron radio emission from an
oscillating magnetic cylinder (with standing sausage waves). It was found that
even for the simple model considered (a straight cylinder), calculating the
emission requires 3D simulations that also take into account the spatial
structure of the MHD waves, because the results can differ from simple
estimates for a spatially homogeneous source (cf. Reznikova et al., 2007;
Mossessian and Fleishman, 2012). At the same time, another complication arises
from the fact that the solar flaring loops are (in contrast to the model of
Reznikova, Antolin, and Van Doorsselaere (2014)) evidently curved. This work
is dedicated to simulating the radio emission from such curved loops.

Simulation of MHD waves in flaring loops is a nontrivial task. Oscillation
modes of a straight overdense cylinder of infinite length are well studied
(Edwin and Roberts, 1983; Nakariakov and Verwichte, 2005, etc.), and the
numerical simulations can be greatly simplified by using the analytical
results. However, these results are not applicable to curved loops (Van
Doorsselaere et al., 2004). Variation of the plasma and magnetic field
parameters as well as of the loop radius with height presents even more
difficulties (Andries et al., 2005; Andries and Cally, 2011). Therefore in
this work we use a simplified model in which an oscillating cylinder is
artificially bent to form a semi-circular loop without actually using the wave
relations in a semi-torus; the magnetic field, plasma density and temperature,
and the loop radius (in the equilibrium state, i.e., without oscillations) are
assumed to be constant along the loop. The main aim of this model is to study
the effect of varying the viewing angle (i.e., the angle between the local
magnetic field and the line of sight) along the loop. Although such a model
cannot correctly describe the entire loop, it can reproduce the effects of the
MHD oscillations in local parts of the loop, where the effects of the loop
curvature and inhomogeneity on the MHD wave parameters are negligible. We
consider the incoherent gyrosynchrotron radiation of energetic electrons which
makes the main contribution to the flare emission in the microwave range.

The simulation model is described in Section model. The simulation results are
presented and discussed in Section results. The conclusions are drawn in
Section conclusion.

##  2 Model

model

###  2.1 MHD Waves

We use the model described earlier by Zaitsev and Stepanov (1975), Edwin and
Roberts (1983), Antolin and Van Doorsselaere (2013), etc. In this model, there
is an overdense cylinder with the magnetic field directed along its axis. The
plasma density and temperature inside the cylinder are higher than outside,
while the magnetic field strength inside the cylinder is lower than outside,
to maintain the pressure balance. We consider the azimuthally symmetric case
when all parameters depend only on the distance along the cylinder axis $z$
and the distance from this axis $r$. In an equilibrium case, the magnetic
field, plasma density, and temperature are assumed to be constant inside and
outside the cylinder; they experience a sharp jump at the cylinder boundary
(at $r=R$).

The formulae describing the MHD disturbances are given, e.g., in the works of
Zaitsev and Stepanov (1975), Edwin and Roberts (1983), Antolin and Van
Doorsselaere (2013), and Reznikova, Antolin, and Van Doorsselaere (2014); see
especially the paper of Reznikova, Antolin, and Van Doorsselaere (2014) where
the solutions of those equations are presented for the same conditions as used
in this work. Therefore we describe the properties of the MHD waves only
briefly. We consider the sausage mode – the compressible azimuthally symmetric
($m=0$) oscillations that affect both the magnetic field and plasma density
and temperature. We consider both the standing and propagating waves which are
described by the relations

$A_{\mathrm{stand}}\propto\sin(\omega t)\sin(kz+\varphi_{A}),\quad
A_{\mathrm{prop}}\propto\sin(\omega t-kz+\varphi_{A}),$ (1)

respectively, where $A$ is the wave amplitude (e.g., in velocity), $\omega$
and $k$ are the wave frequency and wavenumber, respectively, and $\varphi_{A}$
is the initial phase.

Figure 1.: Profiles of the magnetic field strength (a) and direction relative
to the tube axis (b) in the oscillating magnetic tube at different distances
from the tube axis. The standing wave, low-density model (see Table TabMHD) is
considered; one wavelength is shown. Profiles of the plasma density and
temperature are similar to those of the magnetic field.FigMHD

Figure FigMHD presents an example of how the parameters vary with the
coordinates in a standing wave (the oscillation phase is chosen to provide the
maximum deviation from equilibrium). In general, the axial magnetic field
$B_{z}$, plasma density $n_{0}$, and temperature $T_{0}$ vary in phase and
their variations are maximal at the cylinder axis (we are interested only in
the parameters inside the cylinder, because, as shown below, this is the
region where the radio emission is assumed to be generated). Variations of the
above parameters decrease with increasing $r$ and become zero at some radial
distance $r_{*}$ nearby the cylinder boundary ($r_{*}\simeq 0.9R$ for the
parameters considered here). Then the amplitude of the variations increases
slightly, but they are now in anti-phase with the variations of the same
parameters in the inner part of the cylinder. Variations of the radial
magnetic field $B_{r}$ are shifted by $1/4$ period with respect to other
parameters; they are zero at the cylinder axis and reach their maximum at
$r=r_{*}$ (without the phase reversal at that radial distance). Since we
consider low-amplitude oscillations, we have $|B_{r}|\ll|B_{z}|$ (we remind
that $B_{z}$ includes here both the equilibrium magnetic field $B_{0}$ and the
perturbed field component $\delta B_{z}$, i.e., $B_{z}=B_{0}+\delta B_{z}$).
The radial component $B_{r}=\delta B_{r}$ makes almost no contribution to the
resulting magnetic field strength; however, it affects slightly the magnetic
field direction causing it to deviate from the cylinder axis. Variations of
the cylinder radius due to the MHD oscillations are very small and do not
affect the radio emission.

The propagating sausage wave looks like the structure shown in Figure FigMHD
moving along the $z$ axis. All other properties of this wave (including the
phase relations between different parameters) are the same as for the standing
wave.

Table 1.: Parameters of the MHD models: a) cylinder radius ($R$), magnetic fields ($B$), plasma densities ($n_{0}$), and temperatures ($T_{0}$) inside (“in”) and outside (“out”) the cylinder in the equilibrium state, concentration of the energetic electrons ($n_{\mathrm{b}}$) inside the cylinder in the equilibrium state; b) wavelength ($\lambda$) and period ($P$) of the sausage mode, maximum deviations of the magnetic field strength ($\delta B$), magnetic field direction ($\delta\theta$), plasma density ($\delta n_{0}$), plasma temperature ($\delta T_{0}$), and concentration of the energetic electrons ($\delta n_{\mathrm{b}}$) inside the cylinder from the respective equilibrium values.TabMHD Model | Low-density | High-density
---|---|---
Equilibrium cylinder parameters
$R$ [km] | 1000 | 1000
$B_{\mathrm{in}}$ [G] | 160 | 50
$B_{\mathrm{out}}$ [G] | 161 | 56
$n_{0\,\mathrm{in}}$ [$\textrm{cm}^{-3}$] | $4\times 10^{9}$ | $10^{10}$
$n_{0\,\mathrm{out}}$ [$\textrm{cm}^{-3}$] | $10^{9}$ | $3\times 10^{9}$
$T_{0\,\mathrm{in}}$ [K] | $10^{7}$ | $10^{7}$
$T_{0\,\mathrm{out}}$ [K] | $2\times 10^{6}$ | $2\times 10^{6}$
$n_{\mathrm{b}\,\mathrm{in}}$ [$\textrm{cm}^{-3}$] | $10^{8}$ | $2.5\times 10^{8}$
Wave parameters
$\lambda$ [km] | 2825 | 2825
$P$ [s] | 0.35 | 1.5
$(\delta B_{\mathrm{in}})_{\max}$ [G] | $4.05$ | $2.78$
$(\delta\theta_{\mathrm{in}})_{\max}$ | $0.69^{\circ}$ | $1.52^{\circ}$
$(\delta n_{0\,\mathrm{in}})_{\max}$ [$\textrm{cm}^{-3}$] | $1.0\times 10^{8}$ | $6.0\times 10^{8}$
$(\delta T_{0\,\mathrm{in}})_{\max}$ [K] | $1.7\times 10^{5}$ | $4.1\times 10^{5}$
$(\delta n_{\mathrm{b}\,\mathrm{in}})_{\max}$ [$\textrm{cm}^{-3}$] | $2.5\times 10^{6}$ | $1.5\times 10^{7}$

We consider two sets of model parameters, the same as have been used in the
work of Reznikova, Antolin, and Van Doorsselaere (2014); the parameters are
listed in Table TabMHD. The parameters were chosen to provide the same
wavelength $\lambda$ for the given cylinder radius $R$. In the calculations
below, we use mainly the low-density model because those conditions are more
favorable for generation of the gyrosynchrotron radio emission. In the high-
density model, the emission intensity is usually lower; at low frequencies,
the emission becomes strongly affected by the Razin effect (see Section
razin).

###  2.2 Source Geometry

As stated before, the aim of this work is to study the effect of the loop
curvature on the radio emission. For this, we have created model semi-circular
(semi-toroidal) loops by bending the above-mentioned overdense cylinder (see
Figure FigLoop). In this case, the longitudinal coordinate within the cylinder
$z$ is transformed into the distance along the loop axis, and the radial
coordinate $r$ – into the distance from the loop axis (the explicit formulae
are given in the Appendix); this allows us to obtain the plasma and magnetic
field parameters at any given point. Since the lengths of the solar flaring
loops are usually much larger than their radii, we have created the models by
stacking several MHD simulation blocks (each corresponding to one wavelength,
like in Figure FigMHD) along the loop. As the “basic” model, we have used the
loop containing three wavelengths; such loop has the length of 8475 km and the
height (i.e., the curvature radius) of about 2700 km.

The loop orientation is described in general by three Euler angles. However,
since rotation around the line of sight does not provide us with any new
information (it results simply in the same rotation of the simulated radio
images), it is sufficient to consider the variation of only two angles
(Kuznetsov, Nita, and Fleishman, 2011). In this work we assume that the loop
is vertical and located at the solar equator. The loop orientation is
described by two angles: the angle $\psi$ between the loop plane and the
equatorial plane and the longitude $\lambda$. A possible loop inclination is
not considered for the same reason as described above: for a vertical loop,
the variation of the angles $\psi$ and $\lambda$ is sufficient to obtain any
possible loop orientation with respect to the observer (with an accuracy up to
rotation around the line of sight); therefore our results are applicable to
inclined loops as well. However, we note that the loop locations at the solar
disk which are discussed below should be treated as approximate
(illustrative), because they correspond to the zero inclination only.

Figure 2.: Three orientations of the model magnetic loop considered in the
paper: 3D views of the loop (top row); profiles of the viewing angle (relative
to the magnetic field, in the equilibrium state) along the loop axis (middle
row); simulated profiles of the radio intensity in the equilibrium state at
different frequencies (for the low-density model and the electron spectral
index of $\delta=4$) along the loop axis (bottom row). The visible coordinate
along the loop $s$ is measured from the upper (as shown in the 3D views)
footpoint.FigLoop

It is not possible to present here the results for all possible orientations;
therefore we have focused on three illustrative cases (see Figure FigLoop and
Table TabLoops):

a) The loop is located near the disk center. In this case, the viewing angle
$\theta$ (relative to the magnetic field) varies in a wide range along the
loop, being close to zero (or $180^{\circ}$) in the footpoints and being equal
to $90^{\circ}$ at some point near the loop top (see Figure FigLoop). The
coordinate $s$ in Figure FigLoop is the visible distance along the loop axis,
i.e., it takes into account the projection effects.

b) The loop is located near the solar limb and its plane is almost parallel to
the equatorial plane (“Limb A” orientation). In this case, in contrast to the
previous one, the viewing angle is close to $90^{\circ}$ in the loop
footpoints and becomes very small near the loop top.

c) The loop is located near the solar limb and its plane is almost
perpendicular to the line of sight (“Limb B” orientation). In this case the
viewing angle is always close to $90^{\circ}$ and weakly varies along the loop
(although it is not constant, which makes this model different from those
studied in the work of Reznikova, Antolin, and Van Doorsselaere (2014)).

Table 2.: Orientations of the magnetic loop used in the simulations: the angle relative to the equatorial plane ($\psi$) and the longitude ($\lambda$).TabLoops Orientation | Disk center | Limb A | Limb B
---|---|---|---
$\psi$ | $60^{\circ}$ | $20^{\circ}$ | $60^{\circ}$
$\lambda$ | $20^{\circ}$ | $80^{\circ}$ | $80^{\circ}$

###  2.3 Electron Parameters

We assume that the gyrosynchrotron radio emission is produced by the energetic
electrons with the power-law distribution over energy, $f(E)\propto
E^{-\delta}$; in the calculations, the energy range is taken to be
$\textrm{100 keV}<E<\textrm{10 MeV}$ which is typical for the solar flares.
The electrons with lower energies are also present in flares, but their
contribution to the gyrosynchrotron emission is minor (mostly in the form of
self-absorption at low frequencies; see, e.g., Holman (2003)) and usually can
be ignored. The pitch-angle distribution of the electrons is assumed to be
isotropic (i.e., we assume that it is not affected significantly by the MHD
waves). Reznikova, Antolin, and Van Doorsselaere (2014) have considered the
case when the concentration of the energetic electrons $n_{\mathrm{b}}$ inside
the cylinder is proportional to the thermal plasma density,
$n_{\mathrm{b\,in}}/n_{0\,\mathrm{in}}=\textrm{const}$. Alternatively, one may
consider the model in which the energetic electrons are “frozen” into the
magnetic field, so that their concentration is proportional to the field
strength, $n_{\mathrm{b\,in}}/B_{\mathrm{in}}=\textrm{const}$. However, since
in the sausage waves the dominant longitudinal component of the magnetic field
$B_{z}$ and the plasma density $n_{0}$ oscillate in phase, both approaches
provide almost identical results. Therefore in the calculations to follow we
assume that the concentration of the energetic electrons in the loop is always
proportional to the thermal plasma density and
$n_{\mathrm{b\,in}}/n_{0\,\mathrm{in}}=0.025$; in particular, in the
equilibrium state we have $n_{\mathrm{b\,in}}=10^{8}$ and $2.5\times 10^{8}$
$\textrm{cm}^{-3}$ for the low- and high-density models, respectively. We
assume that there are no energetic electrons outside the loop
($n_{\mathrm{b\,out}}=0$).

The gyrosynchrotron radio emission is calculated using the Fast
Gyrosynchrotron Codes (Fleishman and Kuznetsov, 2010; Kuznetsov, Nita, and
Fleishman, 2011; Nita et al., 2015); these codes include the contribution of
the thermal free-free emission as well. We neglect the possible harmonic
structure of the gyrosynchrotron emission at low frequencies by using the
“continuous” regime of the fast gyrosynchrotron codes, because in real
observations this structure should be usually smoothed by the source
inhomogeneity (Kuznetsov, Nita, and Fleishman, 2011); using the “continuous”
approximation also speeds up the calculations appreciably. The codes calculate
the emission characteristics by numerical integration of the radiative
transfer equations for the ordinary and extraordinary electromagnetic modes
while taking into account the possible mode-coupling effects, i.e.,
interaction between the modes in the regions of transverse magnetic field
(Cohen, 1960; Zheleznyakov and Zlotnik, 1964). This approach is sufficient to
reproduce accurately the emission intensity in both the quasi-longitudinal and
quasi-transverse (with respect to the magnetic field) propagation regimes as
well as the circular polarization in the quasi-longitudinal regime (Nita et
al., 2015), although simulating the polarization in the regions with
$\theta\simeq 90^{\circ}$ may require considering the full set of Stokes
parameters (see the upcoming paper of Reznikova, Van Doorsselaere, and
Kuznetsov (2015)). Only the emission and absorption processes inside the
curved magnetic loop are considered, i.e., the emission outside the loop is
assumed to propagate like in vacuum.

Figure 3.: Total (spatially integrated) radio spectra in the equilibrium
state. Solid lines: low-density model; dashed lines: high-density model. Two
electron spectral indices ($\delta=4$ and 6) and the “Limb B” orientation are
considered.FigSpectrum

##  3 Results

results

###  3.1 Equilibrium State

Firstly, we present a brief overview of the emission produced by the loop in
the equilibrium state (i.e., without MHD oscillations). This emission is very
similar to the emission from an oscillating loop averaged over the wave
period. Figure FigSpectrum demonstrates the total (spatially integrated)
intensity spectra for the loop located at the limb (“Limb B” orientation). For
other orientations, the intensity is slightly lower, but the spectra shapes
are qualitatively the same. The concentration of the energetic electrons was
chosen to provide that the spectral maximum (for the low-density model and the
spectral index of the energetic electrons of $\delta=4$) is located at about
10 GHz – a typical value for the continuum microwave emission of solar flares.

The bottom row in Figure FigLoop demonstrates the simulated distributions of
the gyrosynchrotron radio intensity in the equilibrium state along the loop
(for the low-density model). The intensity distributions depend on the
emission frequency and loop orientation: for the loop located near the disk
center, the optically thin emission (e.g., at ${\approx 25}$ GHz) is sharply
peaked near the loop top; the same effect is seen for the emission near the
spectral peak (at ${\approx 10}$ GHz), while the optically thick emission
(e.g., at ${\approx 4}$ GHz) is weakly peaked at the loop footpoints. For the
“Limb A” orientation, the situation is opposite: now the optically thick
emission is peaked near the loop top, while the optically thick emission and
the emission near the spectral peak are peaked at the loop footpoints. A
similar pattern, although with weaker variations, can be seen in the “Limb B”
case. In all cases, there are small narrow peaks/dips at the points where the
local viewing angle equals $90^{\circ}$; they are caused by different
polarization characteristics of the ordinary and extraordinary electromagnetic
modes and are typical of the gyrosynchrotron radiation.

###  3.2 Intensity Modulation: 2D Images

Oscillation-induced variations of the radio emission in different parts of the
magnetic loop tend to compensate each other, thus reducing the variations of
the total (spatially unresolved) emission. Due to the variation of the viewing
angle (and hence of the local emission parameters) along the loop, the
mentioned compensation is not complete. Nevertheless, the variations of the
total intensity are always much weaker than those of the local (spatially
resolved) emission; amplitude of the total intensity variations decreases with
a decrease of the relative wavelength of the MHD waves (i.e., with an increase
of the number of the waves in the loop). Therefore we focus below on the
characteristics of the spatially-resolved emission.

Figure FigImages demonstrates the simulated radio images at the frequency of
25 GHz (optically thin emission) for the standing wave. Two opposite
oscillation phases are shown. To make the effect of the MHD waves more
pronounced, the intensity variations were enhanced as

$I\to\left<I\right>+5(I-\left<I\right>),$ (2)

where $\left<I\right>$ is the intensity averaged over the wave period $P$:

$\left<I\right>=\frac{1}{P}\int_{0}^{P}I(t)\,\mathrm{d}t.$ (3)

We can see in the figure that for the “Limb B” orientation (with an almost
constant viewing angle) the visible emission structure, in general, reflects
the structure of the MHD wave: there are three intensity maxima corresponding
to the points of maximum magnetic field (and hence the maximum concentration
of energetic electrons). For other orientations, due to the projection
effects, the intensity maxima/minima slightly deviate from the magnetic field
maxima/minima; also, not all MHD waves are visible (i.e., both the intensity
and the oscillation-induced variation amplitude depend strongly on the
coordinate along the loop). For the propagating MHD waves, the radio images
are similar to those shown in Figure FigImages, but the brighter/darker spots
drift gradually along the loop with time.

###  3.3 Phase Relations and Modulation Depth

phase We are interested primarily in the qualitative effects that could reveal
the basic MHD wave characteristics (such as the wave mode). As mentioned
above, in real flaring loops the MHD wave properties (including the phase
speed) can vary strongly along the loop, which is not considered in our model.
Therefore it makes no sense to compare, e.g., the simulated time profiles of
the emission at the loop top and the footpoints. Instead we consider the local
emission parameters, i.e., the temporal delays (phase differences) between the
emissions at different frequencies at a given point.

Figures FigTDs-FigTDp show the time-distance plots of the normalized intensity
variation $(I-\left<I\right>)/\left<I\right>$ at the axis of the magnetic loop
(i.e., along the line formed by the projection of the loop axis on the image
plane) for the low-density model. Four frequencies are shown, representing the
optically thick case (4 GHz), “moderately” optically thick case ($6.3$ GHz),
spectral peak (10 GHz), and optically thin case (25 GHz). Above each panel,
the respective maximum modulation depth $\Delta$ is given; this value is
defined as

$\Delta=\max\left|\frac{I-\left<I\right>}{\left<I\right>}\right|.$ (4)

Note that the maximum modulation depth $\Delta$ is different from the average
modulation depth $\epsilon$ given by Equation (epsilon) (which will be used in
Section index). According to Figures FigTDs-FigTDp, the maximum modulation
depth $\Delta$ varies from ${\lesssim 1\%}$ at low frequencies to ${\gtrsim
5\%}$ at high frequencies. These values are relatively low and, in fact, it
would not be easy to detect such oscillations in real observations. However,
this is because we consider the relatively low-amplitude MHD waves: the
variations of the emission source parameters $\delta B/\left<B\right>$ and
$\delta n_{\mathrm{b}}/\left<n_{\mathrm{b}}\right>$ do not exceed $2.5\%$ and
thus the variations of the intensity are comparable to the MHD wave amplitude.
The quasi-periodic pulsations reported by Nakariakov, Melnikov, and Reznikova
(2003), Melnikov et al. (2005), and Reznikova et al. (2007) (with the
modulation depths of up to ${\approx 0.3}$) require much more powerful MHD
waves; nevertheless, if those pulsations are caused by sausage MHD waves,
their characteristics should be qualitatively similar to the case studied in
this work. Note also that the intensity variations are more pronounced in the
optically thin frequency range, in agreement with the results of Mossessian
and Fleishman (2012) and Reznikova, Antolin, and Van Doorsselaere (2014).

Figure FigTDs demonstrates the simulation results for the standing wave. We
can see that the relative modulation depth, in general (excluding the regions
with $\theta\simeq 90^{\circ}$), follows the same trends as the unperturbed
intensity (see Figure FigLoop): for the loop near the disk center, the
modulation depth increases towards the loop top for the optically thin
emission (${\gtrsim 10}$ GHz) and increases towards the loop footpoints for
the optically thick emission (${<10}$ GHz); for the loops near the limb, the
trends are opposite. Oscillations at low ($\lesssim 4$ GHz, the first row of
the figure) and high (${\gtrsim 10}$ GHz, third and fourth rows) frequencies
are synchronous. The fact that the optically thin and thick emissions
oscillate in phase agrees with the results of Reznikova, Antolin, and Van
Doorsselaere (2014), but contradicts the conclusions of Mossessian and
Fleishman (2012). The reason is that, in contrast to Mossessian and Fleishman
(2012), we take into account the modulation of the viewing angle by the
sausage wave but neglect the modulation of the energy of the nonthermal
electrons. In addition, we use the spatially inhomogeneous model in which the
source parameters vary along the line of sight; this is especially important
for the optically thick emission. Since two different approaches (ours and
that of Mossessian and Fleishman (2012)) provide opposite results, we conclude
that the sausage-mode MHD waves (at least, in application to the radio
intensity modulation) cannot be treated without a spatially resolved 3D model.
Note that this behaviour (in-phase variations of the optically thin and thick
emissions) is observed for all loop orientations and at all locations along
the loop; i.e., it is independent of the viewing angle (except the
$\theta=90^{\circ}$ case which is discussed below).

An interesting feature is observed at “intermediate” frequencies (${\approx
6}$ GHz, the second row of Figure FigTDs), i.e., in the optically thick range,
but not far from the spectral peak; the optical depth $\tau$ here exceeds
unity, but is not very large (${\tau\approx 10}$). We can see that the
intensity oscillations at these frequencies are shifted by about $1/4$
wavelength with respect to the oscillations at higher and lower frequencies.
This is caused by the wave-induced variations of the viewing angle: at the
intermediate frequencies, these variations (albeit very small) become the main
factor modulating the intensity. In contrast, at higher or lower frequencies
(with ${\tau<1}$ or ${\tau\gg 1}$), the effect of changing the magnetic field
and electron concentration dominates. At the same time, the modulation depth
at the intermediate frequencies reaches its minimum (in comparison with other
frequency ranges).

The oscillating magnetic field direction (and hence the oscillating viewing
angle) also produces some peculiarities in the parts of the loop where the
viewing angle is close to $90^{\circ}$. Changing the sign of the magnetic
field projection on the line of sight results in changing the dominating
electromagnetic wave mode which affects the intensity. As a result, in the
optically thick frequency range the MHD wave induces intensity variations of
opposite signs at different sides of the $\theta=90^{\circ}$ stripe; the
modulation depth increases sharply. In the optically thin frequency range the
intensity variations at different sides of the $\theta=90^{\circ}$ stripe have
the same sign and the modulation depth demonstrates only minor variations. It
should be noted, however, that the mentioned peculiarities occur only in very
narrow stripes and thus their detection in real observations requires very
high spatial resolution.

Similar peculiarities (although without significant changes of the modulation
depth) can be seen for the “Limb A” orientation, in the optically thick
frequency range, at the distance of about $s\simeq 2$” from the upper
footpoint. They occur due to the specific loop topology: the optically thick
emission is produced near the loop boundary and, for the “Limb A” orientation
(see Figure FigLoop), the dependence of the viewing angle at the loop boundary
surface on the coordinate $s$ can become discontinuous; for other loop
orientations, this effect is not observed.

Figure FigTDp demonstrates the simulation results for the propagating wave.
Since the structure of the standing and propagating sausage waves is the same,
all the above conclusions about the radio intensity oscillations remain valid;
the maximum modulation depths shown in Figure FigTDp are slightly higher than
in Figure FigTDs. Using Figure FigTDp, we can track propagation of the MHD
waves and estimate their speed. For the low-density model considered here (see
Table TabMHD), the phase speed of the sausage waves equals 8070 km
$\textrm{s}^{-1}$; the visible propagation speed of the radio
brightenings/darkenings for the “Limb B” orientation is very close to this
value. However, due to the varying viewing angle, the projected (visible) wave
speed varies along the loop; this effect is especially important for the “Disk
center” and “Limb A” orientations. The amplitude of the propagating radio
intensity variations also varies along the loop with the most notable effects
occurring near the regions of transverse (with respect to the line of sight)
magnetic field: in these regions, the bright/dark stripes in Figure FigTDp
become discontinuous, although the MHD wave itself has no peculiarities; the
bright/dark stripes in the time-distance plots (in the optically thick
frequency range) can also become discontinuous in the regions where the
viewing angle at the loop boundary sharply depends on the coordinates in the
image plane. Thus when interpreting the radio observations with quasi-periodic
pulsations, a special attention should be paid to the $\theta=90^{\circ}$
layers and/or the regions where the visible loop boundary demonstrates
topological peculiarities, since they can introduce additional phase shifts
which do not actually reflect the properties of the underlying MHD waves.

Assuming that the relative intensity variations are approximately proportional
to the relative MHD wave amplitude, we can estimate that the MHD waves with
the amplitude $\delta B/\left<B\right>$ of about 40-50% (with other source
parameters being the same as in Figures FigTDs-FigTDp) should produce the
intensity modulation depth of up to 100%. However, simulating the MHD waves
with such amplitudes requires taking into account the nonlinear effects. As
will be shown in Sections index-razin, the emission modulation depth becomes
higher for softer electron beams or for higher plasma densities at low
frequencies (when the Razin effect is important), although this increase of
the modulation depth is always accompanied by a sharp decrease of the
intensity.

###  3.4 Parametric Study

We now consider how the fluctuations of the radio emission depend on various
parameters of the flaring loop.

####  3.4.1 Effect of the Wave Number

Figure FigWp demonstrates the time-distance plots of the intensity variation
for the models with increased numbers of MHD waves in the magnetic loop (four
and six waves, which corresponds to the loop lengths of 11 300 and 16 950 km,
respectively); the simulation results shown correspond to the low-density
model, nonthermal electrons spectral index of $\delta=4$, and the optically
thin emission (at the frequency of 25 GHz). As already shown above, the
standing and propagating MHD waves affect the radio emission in a
qualitatively similar way; therefore only the results for the propagating wave
are shown.

Our simulations have revealed no significant effect of the number of the MHD
waves in the loop, i.e., both the emission modulation depth and the phase
relations between the emissions at different frequencies remain the same. The
only difference can be noticed in the regions where the loop axis becomes
nearly parallel to the line of sight, i.e., near the loop footpoints for the
“Disk center” orientation and near the loop top for the “Limb A” orientation:
since the length of the formation region of the optically thin emission can
exceed an MHD wavelength, the effects of variation of the magnetic field and
electron concentration in different parts of the emission source (with
opposite phases) can compensate each other. As a result, the amplitude of the
intensity oscillations decreases with a decrease of the relative (with respect
to the loop length) MHD wavelength. This effect is not observed in the
optically thick frequency range.

It can be seen in Figure FigWp (as well as in Figures FigImages-FigTDp) that
the MHD observations are best resolved when the magnetic tube is observed in a
nearly perpendicular direction. Indeed, for the “Limb B” orientation we can
see one-to-one correspondence between the MHD waves and the spatial and
temporal structure of the radio emission. For other orientations, this
correspondence is less straightforward and some MHD wave nodes can even be
missing in the radio emission; as expected, the radio emission cannot
accurately reflect the MHD wave structure in the regions where the viewing
angle with respect to the magnetic field $\theta$ is small. While it is not
easy to estimate the number of observed MHD wave nodes in a general case, we
can expect (and this is confirmed by the simulations) that the MHD wave nodes
become missing in the radio emission in the regions with $\tan\theta\lesssim
R/\lambda$, where $R$ is the loop radius and $\lambda$ is the MHD wavelength;
this happens either due to the averaging along the line of sight (in the
optically thin frequency range) or due to the strong reabsorption of the
emission during propagation (in the optically thick frequency range). Note
also that, as has been discussed above, the $\theta=90^{\circ}$ layers can
introduce the patterns in the radio emission (at low frequencies) similar to
additional MHD waves.

####  3.4.2 Effect of the Spectral Index of Energetic Electrons

index The energy spectrum of the radio-emitting electrons affects greatly the
gyrosynchrotron emission. As can be seen from Figure FigSpectrum, softer
electron beams (with larger spectral indices $\delta$) produce less intense
emission with steeper spectrum slope in the optically thin frequency range;
the spectral peak shifts to lower frequencies. We have found that variation of
the spectral index $\delta$ does not change the simulated radio images (like
in Figure FigImages) qualitatively. However, there are quantitative changes in
the emission time profiles which may affect detectability of the MHD
oscillations.

Figure 8.: Average modulation depth of the intensity at the loop axis for
different frequencies and different spectral indices of the energetic
electrons. The low-density model (propagating wave) and the “Limb B”
orientation are considered.FigIndex

Figure FigIndex demonstrates the average modulation depths of the intensity
(for the propagating MHD wave, three sausage waves per loop, low-density
model, and “Limb B” orientation); the emission at the loop axis is considered
and the values are plotted as functions of the coordinate along the loop. The
average modulation depth $\epsilon$ is calculated as (Mossessian and
Fleishman, 2012)

$\ilabel{epsilon}\epsilon=\frac{m}{\left<I\right>},\qquad
m^{2}=\frac{1}{P}\int_{0}^{P}\left[I(t)-\left<I\right>\right]^{2}\mathrm{d}t;$
(5)

it is related to the maximum modulation depth $\Delta$ (shown in Figures
FigTDs-FigWp) as $\Delta\simeq\epsilon\sqrt{2}$. We can see that in the
optically thin frequency range (Figure FigIndexb) an increase in the spectral
index results firstly in an increase in the modulation depth. As a result, the
oscillation-modulated radio images become more contrasted and the periodic
variations of the emission become easier to detect. For even higher electron
spectral indices (${\delta\gtrsim 10}$ for the considered frequency of 25
GHz), the modulation depth decreases sharply, because the gyrosynchrotron
emission becomes too weak and the emission is produced mainly by the free-free
mechanism that is weakly affected by the MHD oscillations. In the optically
thick frequency range (Figure FigIndexa) the modulation depths are low
(${\lesssim 0.5\%}$) and decrease with increasing $\delta$.

As mentioned above, Figure FigIndex corresponds to the propagating wave and
“Limb B” orientation; for other orientations and/or for the standing wave, the
above conclusions remain the same. Assuming that the relative intensity
variations are approximately proportional to the relative MHD wave amplitude,
we can estimate that for the electron spectral index of $\delta=8$ and the MHD
wave amplitude $\delta B/\left<B\right>$ of about 25-30%, the average
($\epsilon$) and maximum ($\Delta$) modulation depths in the optically thin
frequency range should reach ${\approx 70\%}$ and ${\approx 100\%}$,
respectively; the MHD waves with such amplitudes seem to be feasible.

####  3.4.3 Effect of the Plasma Density

razin Finally, we consider the case when the spectral peak of the
gyrosynchrotron radio emission is formed by the Razin effect (Razin, 1960b,
a). This effect takes place in a relatively high-density plasma and results in
a considerable suppression of the gyrosynchrotron emission at the frequencies
below

$f_{\mathrm{R}}\simeq\frac{2}{3}\frac{f_{\mathrm{p}}^{2}}{f_{\mathrm{B}}},$
(6)

where $f_{\mathrm{p}}$ and $f_{\mathrm{B}}$ are the plasma and cyclotron
frequencies in the emission source, respectively. At the frequencies below
$f_{\mathrm{R}}$, the variations of the thermal plasma density should become
the main factor affecting the radio intensity.

As an illustrative example, we use the high-density model (see Table TabMHD).
As can be seen from Figure FigSpectrum, in this case the Razin frequency
$f_{\mathrm{R}}$ is about 4 GHz. Thus at the frequencies well above 4 GHz the
intensity variations should be caused mainly by the oscillations of the
magnetic field and energetic electron concentration (like in the low-density
model), at the frequencies below 4 GHz the intensity variations should be
caused mainly by the Razin effect (i.e., by the oscillating plasma density),
while at the frequencies around the spectral peak (${\approx 4}$ GHz) both
factors should be important.

Figure FigHD demonstrates the simulation results (time-distance plots of the
intensity variation) at three different frequencies for the high-density
model, standing sausage wave, and the nonthermal electrons spectral index of
$\delta=4$. In the optically thin frequency range, the modulation depth of the
intensity is larger than that for the low-density model due to the model
parameters used (the relative MHD wave amplitude for the high-density model is
larger, see Table TabMHD). The modulation depth reaches its minimum at a
frequency below the spectral peak (${\approx 2}$ GHz, depending on the viewing
angle) and then rapidly increases with decreasing frequency; however, as can
be seen in Figure FigSpectrum, the intensity at those frequencies also becomes
very low, so that detection of the intensity oscillations in observations
would be difficult.

It can be seen in Figure FigHD that for the “Limb B” orientation (when the
viewing angle is close to $90^{\circ}$) the radio oscillations in the
optically thin frequency range (${\gtrsim 10}$ GHz) and near the spectral peak
(${\approx 4}$ GHz) are synchronous (in phase), while the low-frequency
emission (${\lesssim 2}$ GHz) oscillates in anti-phase with the emissions at
higher frequencies; the phase reversal occurs (for ${\theta\simeq
90^{\circ}}$) at ${\approx 2}$ GHz. Clearly, this result indicates that at
high frequencies (above and even slightly below the spectral peak) the
intensity variations are caused by the variations of the magnetic field and
nonthermal electrons concentration, while at lower frequencies (${<2}$ GHz)
the Razin effect dominates; these factors affect the emission in opposite
ways.

A similar picture can be seen for two other loop orientations in the regions
where the viewing angle is close to $90^{\circ}$ (i.e., near the loop top for
the “Disk center” orientation and near the footpoints for the “Limb A”
orientation). For smaller viewing angles the picture is more complicated,
since both the modulation depth and the above-mentioned frequency of the
oscillation phase reversal are angle-dependent: the frequency of the phase
reversal increases with a decrease of the viewing angle. As a result, if the
viewing angle varies gradually along the loop, we can see either correlation
or anti-correlation between the oscillations at two given frequencies,
depending on the coordinate $s$.

The above results contradict the conclusions of Mossessian and Fleishman
(2012) who have found that the oscillations of the high- and low-frequency
emissions (in the case when the Razin effect is important) should occur
strictly in phase. As mentioned before (see Section phase), this is because we
consider a slightly different model for the variation of the parameters in the
sausage wave, as well as the spatially inhomogeneous emission source. Our
results also disagree with the conclusions of Reznikova, Antolin, and Van
Doorsselaere (2014) who have found that the oscillations of the high- and low-
frequency emissions occur mostly in phase (although this conclusion was
dependent on the viewing angle). Most likely, this is because Reznikova,
Antolin, and Van Doorsselaere (2014) considered the emission with the
frequencies below the spectral peak but above the “phase reversal frequency”
so that the effect of the Razin suppression was too weak.

##  4 Conclusions

conclusion We have simulated the modulation of the gyrosynchrotron radio
emission by the sausage-mode MHD oscillations in a semi-circular magnetic
loop, in application to the solar flares. Although the model used does not
include the variation of the magnetic field and plasma parameters with height,
it has allowed us to study the effect of varying the viewing angle along the
loop. We have considered three loop orientations and different parameters of
the magnetic field, plasma, and energetic particles. In particular, we have
studied in detail the phase relations between the spatially resolved intensity
oscillations at different frequencies. The results can be summarized as:

  * •

In the model with a relatively low plasma density (when the Razin effect is
negligible), the high-frequency (optically thin) and low-frequency (optically
thick) emissions oscillate, as a rule, in phase. An exception is the optically
thick emission at the frequencies just below the spectral peak, whose
oscillations are shifted by ${\approx 1/4}$ MHD wavelength with respect to the
oscillations at other frequencies.

  * •

In the model with a relatively high plasma density (when the Razin effect is
important), the emissions at high and low frequencies oscillate in anti-phase.
The reversal of the oscillation phase occurs, as a rule, at a frequency well
below the spectral peak; this frequency depends on the viewing angle.

  * •

In the optically thin frequency range, both the unperturbed intensity and the
absolute and relative intensity variations are the highest in the regions
where the local magnetic field is nearly perpendicular to the line of sight.
In contrast, in the optically thick frequency range all the mentioned values
are the highest in the regions with small viewing angles.

  * •

In the optically thin frequency range, the intensity oscillations are more
pronounced for softer electron beams (i.e., with higher spectral indices). In
the optically thick frequency range the trend is opposite.

  * •

In the optically thick frequency range, the intensity oscillations can have
sharp phase shifts between the adjacent regions due to the presence of layers
of transverse (with respect to the line of sight) magnetic field and other
topological peculiarities.

  * •

Averaging over the visible source area and/or along the line of sight reduces
the amplitude of the intensity oscillations; this effect becomes more
important with an increase in the number of the MHD harmonics in the loop.

The latter result means that, due to a limited spatial resolution, the real
observations can differ from the ideal simulations presented here.
Nevertheless, we expect that if an instrument spatial resolution is better
than a MHD wavelength, this will allow us to detect (or rule out) the above-
mentioned phase relations between the oscillations at different frequencies;
another requirement is the ability to perform imaging observations at many
frequencies. We anticipate that the forthcoming multiwavelength
radioheliographs (such as the Expanded Owens Valley Solar Array, Upgraded
Siberian Solar Radio Telescope, and Chinese Spectral Radioheliograph; see the
review of Nakariakov, Kashapova, and Yan (2014) and references therein) will
be able to provide the necessary data.

##### Acknowledgements

This work was supported in part by the Russian Foundation of Basic Research
(grants 12-02-00173, 13-02-90472 and 14-02-91157) and by a Marie Curie
International Research Staff Exchange Scheme “Radiosun”
(PEOPLE-2011-IRSES-295272). The research leading to the presented results has
been sponsored by an Odysseus grant of the FWO Vlaanderen. It has been
performed in the context of the IAP P7/08 CHARM (Belspo) and the GOA-2015-014
(KU Leuven).

##  Appendix A Toroidal Coordinates

toroidal As was mentioned above, we consider the MHD waves in an overdense
cylinder; the system can be described by the cylindrical coordinates: the
distance along the cylinder axis $z$, the distance from this axis $r$, and the
azimuthal angle $\varphi$ (see Figure FigCoordinatesa). We assume that the
cylinder has the length $L$ (i.e., $0\leq z\leq L$) and the radius $R$ (the
radio emission is produced only inside the cylinder, i.e., at $r\leq R$).

The cylinder is assumed to be bent into a semi-circular (semi-toroidal) loop
(see Figure FigCoordinatesb); the loop curvature radius is $H=L/\pi$
(evidently, we need $H>R$). We assume that initially the loop is oriented as
shown in Figure FigCoordinatesb, i.e., it is vertical, symmetric with respect
to the $x^{\prime}z^{\prime}$ and $y^{\prime}z^{\prime}$ planes, and its axial
line lies in the $x^{\prime}z^{\prime}$ plane. Then the cylindrical
coordinates $(z,r,\varphi)$ are related to the Cartesian coordinates
$(x^{\prime},y^{\prime},z^{\prime})$ as

$\displaystyle x^{\prime}$ $\displaystyle=$
$\displaystyle(r\cos\varphi+H)\cos\chi,$ (7) $\displaystyle y^{\prime}$
$\displaystyle=$ $\displaystyle r\sin\varphi,$ (8) $\displaystyle z^{\prime}$
$\displaystyle=$ $\displaystyle(r\cos\varphi+H)\sin\chi,$ (9)

and

$\displaystyle\tan\chi$ $\displaystyle=$
$\displaystyle\frac{z^{\prime}}{x^{\prime}},$ (10) $\displaystyle r^{2}$
$\displaystyle=$
$\displaystyle(x^{\prime}-H\cos\chi)^{2}+(y^{\prime})^{2}+(z^{\prime}-H\sin\chi)^{2},$
(11) $\displaystyle\tan\varphi$ $\displaystyle=$
$\displaystyle\frac{y^{\prime}}{x^{\prime}\cos\chi+z^{\prime}\sin\chi-H},$
(12)

where $\chi=z/L$.

Figure 10.: a) Sketch of the model plasma cylinder. b) Sketch of the model
semi-toroidal loop.FigCoordinates

Assume that a vector $\mathbf{A}$ at a given point is defined by its three
components $(A_{z},A_{r},A_{\varphi})$ in the cylindrical coordinate system.
After transformation to the semi-circular loop, the vector can be expressed as
$\mathbf{A}^{\prime}=A_{z}\mathbf{a}^{\prime}_{z}+A_{r}\mathbf{a}^{\prime}_{r}+A_{\varphi}\mathbf{a}^{\prime}_{\varphi}$,
where the unit vectors $\mathbf{a}^{\prime}_{z,r,\varphi}$ in the Cartesian
coordinates are given by

$\displaystyle\mathbf{a}^{\prime}_{z}$ $\displaystyle=$
$\displaystyle(-\sin\chi,0,\cos\chi),$ (13)
$\displaystyle\mathbf{a}^{\prime}_{r}$ $\displaystyle=$
$\displaystyle(\cos\varphi\cos\chi,\sin\varphi,\cos\varphi\sin\chi),$ (14)
$\displaystyle\mathbf{a}^{\prime}_{\varphi}$ $\displaystyle=$
$\displaystyle(-\sin\varphi\cos\chi,\cos\varphi,-\sin\varphi\sin\chi).$ (15)

The above formulae are valid in a general case. In this work we consider the
azimuthally symmetric case when all parameters are independent of the
azimuthal angle $\varphi$ and the azimuthal components $A_{\varphi}$ of all
vectors are equal to zero.

The orientation shown in Figure FigCoordinatesb corresponds to the vertical
loop located at the center of the solar disk, with the axis $z^{\prime}$ being
the line of sight and the plane $x^{\prime}z^{\prime}$ being the equatorial
plane. All other orientations (see, e.g., Figure FigLoop) are obtained by
rotating the loop firstly by the angle $\psi$ around the local vertical
($z^{\prime}$ axis) and then by the angle $\lambda$ around the solar rotation
axis.

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