Astronomy & Astrophysics manuscript no. 7246 c© ESO 2019
August 20, 2019
Torsional oscillations of longitudinally inhomogeneous coronal
loops
T.V. Zaqarashvili1 & K. Murawski2
1 Georgian National Astrophysical Observatory (Abastumani Astrophysical Observatory), Kazbegi Ave. 2a, Tbilisi 0160, Georgia
2 Group of Astrophysics and Gravity Theory, Institute of Physics, UMCS, ul. Radziszewskiego 10, 20-031 Lublin, Poland
received / accepted
ABSTRACT
Aims. We explore the effect of an inhomogeneous mass density field on frequencies and wave profiles of torsional Alfvén oscillations
in solar coronal loops.
Methods. Dispersion relations for torsional oscillations are derived analytically in limits of weak and strong inhomogeneities. These
analytical results are verified by numerical solutions, which are valid for a wide range of inhomogeneity strength.
Results. It is shown that the inhomogeneous mass density field leads to the reduction of a wave frequency of torsional oscillations, in
comparison to that of estimated from mass density at the loop apex. This frequency reduction results from the decrease of an average
Alfvén speed as far as the inhomogeneous loop is denser at its footpoints. The derived dispersion relations and wave profiles are
important for potential observations of torsional oscillations which result in periodic variations of spectral line widths.
Conclusions. Torsional oscillations offer an additional powerful tool for a development of coronal seismology.
Key words. Magnetohydrodynamics (MHD) – Sun: corona – Sun: oscillations
1. Introduction
Recent space-based observations revealed a presence of vari-
ous kinds of magnetohydrodynamic (MHD) waves and oscilla-
tions in the solar corona. These observations as well as mod-
eling of MHD waves are important as these waves contribute
to the coronal heating problem (Roberts 2000) and they may
consist unique tool of a coronal seismology (Edwin & Roberts
1983, Nakariakov & Ofman 2001). Fast kink (Aschwanden et
al. 1999, Nakariakov et al. 1999, Wang & Solanki 2004) and
sausage (Nakariakov 2003, Pascoe et al. 2007) as well as slow
(de Moortel et al. 2002, Wang et al. 2003) magnetosonic oscil-
lations were observed to be associated either with or without
a solar flare. Analytical studies of these oscillations in coronal
loops were carried on over the last few decades, amongst others,
by Edwin & Roberts (1982, 1983), Poedts & Boynton (1996),
Nakariakov (2003), Van Doorsselaere et al. (2004a,b), Ofman
(2005), Verwichte et al. (2006) and Diáz et al. (2006).
Coronal loops act as natural wave guides for magnetosonic
and torsional Alfvén waves. The later are purely azimuthal os-
cillations in cylindrical geometry. In the linear regime, Alfvén
oscillations do not lead to mass density perturbations. As a re-
sult, contrary to magnetosonic waves, torsional Alfvén waves
can be observed only spectroscopically. While propagating from
the base of the solar corona along open magnetic field lines,
these waves may lead to an increase of a spectral line width with
height (Hassler et al. 1990, Banerjee et al. 1998, Doyle et al.
1998). In closed magnetic field structures, such as coronal loops,
these waves can be observed indirectly as periodic variations of
non-thermal broadening of spectral lines (Zaqarashvili 2003).
Alongside magnetosonic waves, torsional oscillations can be
used to infer, in the framework of coronal seismology, plasma
properties inside oscillating loops. These oscillations are an ideal
Send offprint requests to: T. Zaqarashvili e-mail: temury@genao.org
tool of coronal seismology as their phase speed depends alone on
plasma quantities within the loop, while wave speeds of magne-
tosonic oscillations are influenced by plasma conditions in the
ambient medium. Having known mass density within a loop,
coronal seismology, that is based on torsional oscillations, en-
ables to estimate a magnetic field strength. Torsional oscilla-
tions are potentially important in the context of rapid attenua-
tion of coronal loop kink oscillations (Aschwanden et al. 1999,
Nakariakov et al. 1999). One of a few suggested mechanisms
of the attenuation is a resonant absorption of fast magnetosonic
kink waves by azimuthal Alfvén waves (Ruderman & Roberts
2002). This process may lead to a formation of torsional oscil-
lations in the outer part of a loop. As a result, spotting torsional
oscillations after the kink mode was attenuated would serve as
an evidence of this attenuation mechanism.
A theoretical study of Alfvén oscillations in a coronal loop
was carried on recently by Gruszecki et al. (2007) who con-
sidered impulsively generated oscillations in two-dimensional
straight and curved magnetic field topologies. They found that
lateral leakage of Alfvén waves into the ambient corona is neg-
ligibly small. However, mass density profiles were adopted ho-
mogeneous within the loop, while the real conditions there are
much more complex.
Despite of significant achievements in a development of re-
alistic models there is still much more effort required to develop
our knowledge of wave phenomena in coronal loops. A goal of
this paper is to study the influence of inhomogeneous mass den-
sity fields on spectrum of torsional oscillations. The paper is or-
ganized as follows. Analytical solutions for torsional oscillations
in a longitudinally inhomogeneous coronal loop are presented in
Sect. 2. The numerical results are showed in Sect. 3. Guidelines
for potential observations of these oscillations are presented in
Sect. 4. This paper is concluded by a discussion and a short sum-
mary of the main results in Sect. 5.
http://arxiv.org/abs/0704.0360v1
2 T. Zaqarashvili & K. Murawski: Torsional oscillations of a coronal loop
2. Analytical model of torsional oscillations
We consider a coronal loop of its inhomogeneous mass density
̺0(z) and length 2L, that is embedded in a uniform magnetic field
B = B0ẑ. Small amplitude torsional Alfvén waves in a cylindri-
cal coordinate system (r, φ, z), in which plasma profiles depend
on a longitudinal coordinate z only, can be described by the fol-
lowing linear equations:
4π̺0(z)
, (1)
, (2)
where uφ and bφ are the velocity and magnetic field components
of Alfvén waves.
These equations can be easily cast into a single wave equa-
V2A(z)
= 0 , (3)
where VA(z) = B0/
4π̺0(z) is the Alfvén speed. Assuming that
uφ ∼ exp(iωt), where ω is a wave frequency, we get the equation
uφ = 0 . (4)
For a trapped solution uφ must satisfy line-tying boundary con-
ditions which are implemented by setting
uφ(z = ±L) = 0 . (5)
Equation (4) with condition (5) consists the well-known Sturm-
Liuville problem which solution depends on the profile of VA(z).
We model the coronal loop by a rarefied plasma at the loop apex
(at z = 0) and by a compressed plasma at the loop footpoints
(z = ±L). Specifically, we adopt
̺0(z) = ̺00
1 + α2
, (6)
where ̺00 is the mass density at the loop apex and α
2 is a param-
eter which defines a strength of the inhomogeneity. For α2 = 0
the above mass density profile corresponds to a homogeneous
loop, while for a larger value of α2 the medium is more inhomo-
geneous. Figure 1 illustrates ̺0(z) for α
2 = 50. The mass density
is described by Eq. (6) with ̺00 = 10
−12 kg m−3 and L = 25 Mm.
Note that plasma is compressed at z = ±L. Substituting Eq. (6)
into Eq. (4), we obtain
1 + α2
uφ = 0 , (7)
where VA0 = B0/
4π̺00. With a use of the notation
y ≡ uφ, x ≡
z, a ≡ −
Eq. (7) can be rewritten in the form of Weber (parabolic cylinder)
equation (Abramowitz & Stegun 1964)
y = 0 . (9)
Fig. 1. Spatial profile of the background mass density, ̺0(z), given by
Eq. (6) with α2 = 50. The mass density and length are expressed in
units of 10−12 kg m−3 and 1 Mm, respectively.
Standard solutions to this equation are called Weber (parabolic
cylinder) functions (Abramowitz & Stegun 1964)
W(a,±x) =
(coshπa)1/4
G1y1(x) ∓
2G3y2(x)
, (10)
where
, G3 =
and y1(x), y2(x) are respectively even and odd solutions to
Eq. (9)
y1(x) = 1 + a
+ · · · ,
y2(x) = x + a
+ · · · .
2.1. Two limiting solutions
Periodic solutions to Eq. (9) can be written analytically in the
limiting cases: (a) for a large value of a but a moderate value
of x; (b) for a large x but a moderate a. The first (second) case
corresponds to α2 ≪ 1 (α2 ≫ 1).
2.1.1. Weakly inhomogeneous plasma
We consider first the case of a weakly inhomogeneous mass den-
sity field, i.e. α2 ≪ 1. In this case we have
a < 0, −a≫ x2, p ≡
−a . (12)
We adopt the following expansion (Abramowitz & Stegun
1964):
W(a, x) + iW(a,−x) =
2W(a, 0) exp [vr + i(px + π/4 + vi)] , (13)
where
W(a, 0) =
, (14)
vr = −
(x/2)2
(2p)2
2(x/2)4
(2p)4
+ · · · , vi =
2/3(x/2)3
+ · · · . (15)
T. Zaqarashvili & K. Murawski: Torsional oscillations of a coronal loop 3
As a result of relation −a≫ x2 we have from Eq. (13)
W(a, x) =
2W(a, 0) exp
cos ζ , (16)
W(a,−x) =
2W(a, 0) exp
sin ζ , (17)
ζ ≡ px + π/4 +
. (18)
The general solution to Eq. (9) is
uφ = c1W(a, x) + c2W(a,−x) , (19)
where c1 and c2 are constants.
For a homogeneous loop, i.e. α2 = 0, we recognize the well
known solution
uφ ∼ c1 cos (kz + π/4) + c2 sin (kz + π/4). (20)
Here wave number k satisfies the following homogeneous dis-
persion relation:
. (21)
Line-tying boundary conditions of Eq. (5) lead then to discrete
values of the wave frequency, viz.
1 + α2/6
, n = 1, 2, 3, . . . . (22)
From this dispersion relation we infer that in a comparison to
the loop with a homogeneous mass density distribution, ̺00, the
weakly inhomogeneous mass density field results in a decrease
of a wave frequency. This reduction is a consequence of the fact
that the inhomogeneous loop is denser at its footpoints, so the
average Alfvén speed is decreased. To show this, we compare the
results for the inhomogeneous loop with the homogeneous loop
with the same average density, so that both loops contain exactly
the same mass (Andries et al. 2005). We introduce a frequency
difference
∆ωn = ωn − ω̄n , (23)
where
ω̄n =
V̄A0 =
4π ¯̺0
corresponds to the average mass density
¯̺0 =
̺0(z)dz = ̺00
. (25)
Substituting Eq. (25) into Eq. (24), we obtain
ω̄n =
1 + α2/3
. (26)
From Eqs. (23) and (26) we find that ∆ωn ≤ 0. Here we infer
that in comparison to the average mass density case the wave
frequency is reduced, but as a result of α2 ≪ 1 the frequency
reduction is small. This is in a disagreement with Fermat’s law
and with the results of Murawski et al. (2004) who showed that
sound waves experience frequency increase in a case of a space-
dependent random mass density field.
2.1.2. Strongly inhomogeneous plasma
We discuss now a strongly inhomogeneous mass density case,
i.e. α2 ≫ 1. This case corresponds to x ≫ |a|. In this limit we
get (Abramowitz & Stegun 1964)
W(a, x) =
2k/x(s1(a, x) cos(ξ) − s2(a, x) sin(ξ)) , (27)
W(a,−x) =
2/kx(s1(a, x) sin(ξ) − s2(a, x) cos(ξ)) , (28)
where
− a ln x +
argΓ(1/2 + ia)
, (29)
1 + e2πa − eπa, (30)
s1(a, x) ∼ 1 +
1!2x2
2!22x4
− · · · , (31)
s2(a, x) ∼ −
1!2x2
2!22x4
+ · · · (32)
ur + ivr = Γ(r + 1/2 + ia)/Γ(1/2 + ia), r = 2, 4, . . . . (33)
The boundary conditions of Eq. (5) lead to the discrete fre-
quency spectrum
. (34)
Here we infer that the strongly inhomogeneous mass den-
sity field results in a significant decrease of a wave frequency in
comparison to the case of the loop with the constant density, ̺00.
This wave frequency decrease is a consequence of the fact that
the inhomogeneous loop is denser at its footpoints. Substituting
Eq. (34) into Eq. (23) we find that ∆ωn > 0. This wave frequency
decrease, in a comparison to the case of an average mass density
is now in an agreement with Fermat’s law and with the results of
Murawski et al. (2004).
3. Numerical results
Numerical simulations are performed for Eqs. (1), (2) with an
adaptation of CLAWPACK which is a software package de-
signed to compute numerical solutions to hyperbolic partial dif-
ferential equations using a wave propagation approach (LeVeque
2002). The simulation region (−L, L) is covered by an uniform
grid of 600 numerical cells. We verified by convergence studies
that this grid does not introduce much numerical diffusion and as
a result it represents well the simulation region. We set reflect-
ing boundary conditions at the left and right boundaries of the
simulation region.
Figure 2 shows a spatial profile of velocity uφ(z) for α
2 = 50,
drawn at t = 1000 s (solid line). This spatial profile results from
the initial Gaussian pulse that was launched at t = 0 in the center
of the simulation region, at z = 0. It is noteworthy that the sine-
wave profile of Eq. (20), which is valid for α2 = 0 (dashed line),
is distorted by the strong inhomogeneity which takes place for
the case of α2 = 50.
As a consequence of the inhomogeneity wave period is al-
tered. Figure 3 displays wave period P vs. inhomogeneity param-
eter α2. Diamonds represent the numerical solutions while the
solid lines correspond to the analytical solution to Eqs. (22) (top
panel) and (34) (bottom panel). Wave periods were obtained by
Fourier analysis of the wave signals that were collected in time
at the fixed spatial location, z = 0. It is discernible that the nu-
merical data fits quite well to the analytical curves. A growth of
4 T. Zaqarashvili & K. Murawski: Torsional oscillations of a coronal loop
Fig. 2. Numerically evaluated velocity profile uφ at t = 1000 s for α2 =
50 (solid line). This profile corresponds to the mode number n = 1. Note
that as a result of strong inhomogeneity, uφ departs from the sine-wave
which corresponds to α2 = 0. The dashed line corresponds to Eq. (20)
with c1 = c2 = 0.5.
Fig. 3. Wave period P = ω/2π vs. α2 for the mode number n = 1.
Diamonds correspond to the numerical solutions to Eqs. (1), (2). Solid
lines are drawn with the use of the analytical solution to Eqs. (22) and
(34). The wave period is expressed in seconds.
wave period P with α2 results from wave scattering on centers
of the inhomogeneity and it can be explained on simple physical
grounds. In an inhomogeneous field wave frequency ωn of the
torsional oscillations can be estimated from the following for-
mula:
V̄A0 , (35)
where V̄A0 is the averaged Alfvén speed that is expressed by Eq.
(24). Using P = 2π/ωn we obtain
4π ¯̺0
. (36)
As ¯̺0 grows with α, the growth of P with α results in.
4. Potential observations of torsional oscillations
Torsional oscillations of a coronal loop may result in periodic
variations of spectral line non-thermal broadening (expressed by
a half line width, ∆λB, hereafter HW) (Zaqarashvili 2003). For a
homogeneous loop, HW can be expressed as
∆λB =
uVA0λ
|sin(ωnt)sin(knz)| , (37)
where u is an amplitude of oscillations, λ is a wave length of
the spectral line and c is the light speed. Periodic variations of
spectral line width depend on a height above the solar surface:
a strongest variation corresponds to the wave antinode and the
place of a lack of line width variation corresponds to the nodes
(loop footpoints). Therefore, time series of spectroscopic ob-
servations may allow to determine a wave period. Knowing a
length of the loop, we may estimate the Alfvén speed, which in
turn gives a possibility to infer the magnetic field strength in the
corona. We estimate the expected value of line width variations
which result from torsional oscillations. For a typical coronal
Alfvén speed of ∼ 800 km/s, an amplitude of linear torsional
oscillation can be ∼ 40 km/s, which consists 5% of the Alfvén
speed. For the ”green” coronal line Fe XIV (5303 Å) from Eq.
(37) we obtain
∆λB ≈ 0.7 Å. (38)
This value is about twice larger than the original thermal broad-
ening of Fe XIV line. As a consequence, torsional oscillations
can be detected in time series of the green coronal line spectra.
For a weakly inhomogeneous distribution of mass density
along a loop, Eq. (22) enables to estimate the Alfvén speed at
the loop apex with the help of the observed period of HW vari-
ation and a loop length. For a strongly inhomogeneous density
profile along a loop, Eq. (34) shows that a wave period of tor-
sional oscillations is not just the ratio of the loop length to the
Alfvén speed, but it strongly depends on the rate of inhomo-
geneity, α2. Therefore, an additional effort is required in order
to apply the method of coronal seismology for torsional oscil-
lations. A spatial variation of mass density along the loop can
be estimated by a direct measurement of spectral line intensity
variation along the loop. Then, the estimated variation can be fit-
ted to Eq. (6), and hence a value of α2 can be inferred. Eq. (34)
provides a value of VA0 at the loop summit. Another possibility
is to collect time series of spectroscopic observations at differ-
ent positions of the loop. A spatial variation of line width along
the loop may be compared to the theoretical plot of uφ (Fig. 2),
which enables to estimate α2 and consequently Alfvén speed at
the loop apex (with a use of Eqs. (22) or (34)).
T. Zaqarashvili & K. Murawski: Torsional oscillations of a coronal loop 5
5. Discussion and summary
It is commonly believed that Alfvén waves are generated in the
solar interior either by convection (granulation, supergranula-
tion) or by any other kinds of plasma flow (differential rotation,
solar global oscillations). Due to their incompressible nature,
these waves may carry energy from the solar surface to the solar
corona and therefore they may significantly contribute to coro-
nal heating and solar wind acceleration. In closed magnetic loops
the Alfvén waves may set up the standing torsional oscillations,
while in opened magnetic structures these waves may propagate
up to the solar wind. As a result, observations of Alfvén waves
can be of vital importance to the problems of plasma heating and
particle acceleration.
The Alfvén waves that propagate along open magnetic field
lines may lead to a growth of a spectral line width with height
(Hassler et al. 1990, Banerjee et al. 1998; Doyle et al. 1998).
However, at some altitudes the spectral line width reveals a sud-
den fall off (Harrison et al. 2002; O’Shea et al. 2003, 2005). This
phenomenon was recently explained by resonant energy transfer
into acoustic waves (Zaqarashvili et al. 2006).
On the other hand, the photospheric motions may set up tor-
sional oscillations in closed magnetic loop systems, which can
be observed spectroscopically as periodic variations of spectral
line width (Zaqarashvili 2003). As a result, the observation of
Alfvén waves can be used as an additional powerful tool of coro-
nal seismology; the observed period and loop mean length en-
ables to estimate the Alfvén speed within a loop, which in turn
makes it possible to infer a mean magnetic field strength.
Besides their photospheric origin, torsional Alfvén waves
can be generated in the solar corona in a process of resonant ab-
sorption of the global oscillations (Ruderman & Roberts 2002,
Goossens et al. 2002, Andries et al. 2005, Terradas et al. 2006).
These oscillations may excite Alfvén waves in the outer inho-
mogeneous part of a loop, leading to attenuation of global oscil-
lations and amplification of torsional oscillations. These Alfvén
oscillations can be detected as periodic variations of spectral line
width. As a consequence, observations of Alfvén waves can be
a key for a determination of a damping mechanism of the loop
global oscillations.
Dynamics of torsional Alfvén waves in a homogeneous loop
can be easily solved. However, real coronal loops are longitudi-
nally inhomogeneous, which leads to alteration of wave dynam-
ics (Arregui et al. 2005, 2007, Van Doorsselaere et al. 2004a,b,
Donnelly et al. 2006, Dymova & Ruderman 2006, McEwan et
al. 2006). Therefore, the dynamics of Alfvén waves in longitudi-
nally inhomogeneous coronal loops must be understood in order
to provide analytical basis for potential observations of torsional
oscillations.
In this paper we discussed by analytical and numerical
means evolution of torsional Alfvén waves in an inhomogeneous
mass density field. The analytical efforts resulted in dispersion
relations which were obtained for a specific choice of an equilib-
rium mass density profile. These dispersion relations were writ-
ten explicitly for two limiting cases: (a) weekly inhomogeneous
and (b) strongly inhomogeneous mass density fields. From these
dispersion relations we inferred that the inhomogeneity results in
a wave frequency reduction in comparison to that of estimated at
the loop summit. This analytical finding is supported by the nu-
merical data which reveals that frequency reduction takes place
outside the region of validity of the analytical approach. As a
result of that we claim that a reduction of wave frequency is
ubiquitous for the inhomogeneous mass density field we consid-
ered. This reduction is a consequence of wave scattering on in-
homogeneity centers and it results from reduction of the average
Alfvén speed within a coronal loop. This frequency reduction
has important implications as far as wave observations are con-
cerned. The analytical formulae can be used for estimation of
coronal plasma parameters and therefore torsional Alfvén waves
consist an additional powerful tool of coronal seismology.
Acknowledgments: The authors express their thanks to the
referee, Prof. S. Poedts, for his stimulating comments. The work
of T.Z. is supported by the grant of Georgian National Science
Foundation GNSF/ST06/4-098. A part of this paper is sup-
ported by the ISSI International Programme ”Waves in the Solar
Corona”.
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	Introduction
	Analytical model of torsional oscillations
	Two limiting solutions
	Weakly inhomogeneous plasma
	Strongly inhomogeneous plasma
	Numerical results
	Potential observations of torsional oscillations
	Discussion and summary

  We explore the effect of an inhomogeneous mass density field on frequencies
and wave profiles of torsional Alfven oscillations in solar coronal loops.
Dispersion relations for torsional oscillations are derived analytically in
limits of weak and strong inhomogeneities. These analytical results are
verified by numerical solutions, which are valid for a wide range of
inhomogeneity strength. It is shown that the inhomogeneous mass density field
leads to the reduction of a wave frequency of torsional oscillations, in
comparison to that of estimated from mass density at the loop apex. This
frequency reduction results from the decrease of an average Alfven speed as far
as the inhomogeneous loop is denser at its footpoints. The derived dispersion
relations and wave profiles are important for potential observations of
torsional oscillations which result in periodic variations of spectral line
widths. Torsional oscillations offer an additional powerful tool for a
development of coronal seismology.

Introduction
Recent space-based observations revealed a presence of vari-
ous kinds of magnetohydrodynamic (MHD) waves and oscilla-
tions in the solar corona. These observations as well as mod-
eling of MHD waves are important as these waves contribute
to the coronal heating problem (Roberts 2000) and they may
consist unique tool of a coronal seismology (Edwin & Roberts
1983, Nakariakov & Ofman 2001). Fast kink (Aschwanden et
al. 1999, Nakariakov et al. 1999, Wang & Solanki 2004) and
sausage (Nakariakov 2003, Pascoe et al. 2007) as well as slow
(de Moortel et al. 2002, Wang et al. 2003) magnetosonic oscil-
lations were observed to be associated either with or without
a solar flare. Analytical studies of these oscillations in coronal
loops were carried on over the last few decades, amongst others,
by Edwin & Roberts (1982, 1983), Poedts & Boynton (1996),
Nakariakov (2003), Van Doorsselaere et al. (2004a,b), Ofman
(2005), Verwichte et al. (2006) and Diáz et al. (2006).
Coronal loops act as natural wave guides for magnetosonic
and torsional Alfvén waves. The later are purely azimuthal os-
cillations in cylindrical geometry. In the linear regime, Alfvén
oscillations do not lead to mass density perturbations. As a re-
sult, contrary to magnetosonic waves, torsional Alfvén waves
can be observed only spectroscopically. While propagating from
the base of the solar corona along open magnetic field lines,
these waves may lead to an increase of a spectral line width with
height (Hassler et al. 1990, Banerjee et al. 1998, Doyle et al.
1998). In closed magnetic field structures, such as coronal loops,
these waves can be observed indirectly as periodic variations of
non-thermal broadening of spectral lines (Zaqarashvili 2003).
Alongside magnetosonic waves, torsional oscillations can be
used to infer, in the framework of coronal seismology, plasma
properties inside oscillating loops. These oscillations are an ideal
Send offprint requests to: T. Zaqarashvili e-mail: temury@genao.org
tool of coronal seismology as their phase speed depends alone on
plasma quantities within the loop, while wave speeds of magne-
tosonic oscillations are influenced by plasma conditions in the
ambient medium. Having known mass density within a loop,
coronal seismology, that is based on torsional oscillations, en-
ables to estimate a magnetic field strength. Torsional oscilla-
tions are potentially important in the context of rapid attenua-
tion of coronal loop kink oscillations (Aschwanden et al. 1999,
Nakariakov et al. 1999). One of a few suggested mechanisms
of the attenuation is a resonant absorption of fast magnetosonic
kink waves by azimuthal Alfvén waves (Ruderman & Roberts
2002). This process may lead to a formation of torsional oscil-
lations in the outer part of a loop. As a result, spotting torsional
oscillations after the kink mode was attenuated would serve as
an evidence of this attenuation mechanism.
A theoretical study of Alfvén oscillations in a coronal loop
was carried on recently by Gruszecki et al. (2007) who con-
sidered impulsively generated oscillations in two-dimensional
straight and curved magnetic field topologies. They found that
lateral leakage of Alfvén waves into the ambient corona is neg-
ligibly small. However, mass density profiles were adopted ho-
mogeneous within the loop, while the real conditions there are
much more complex.
Despite of significant achievements in a development of re-
alistic models there is still much more effort required to develop
our knowledge of wave phenomena in coronal loops. A goal of
this paper is to study the influence of inhomogeneous mass den-
sity fields on spectrum of torsional oscillations. The paper is or-
ganized as follows. Analytical solutions for torsional oscillations
in a longitudinally inhomogeneous coronal loop are presented in
Sect. 2. The numerical results are showed in Sect. 3. Guidelines
for potential observations of these oscillations are presented in
Sect. 4. This paper is concluded by a discussion and a short sum-
mary of the main results in Sect. 5.
http://arxiv.org/abs/0704.0360v1
2 T. Zaqarashvili & K. Murawski: Torsional oscillations of a coronal loop
2. Analytical model of torsional oscillations
We consider a coronal loop of its inhomogeneous mass density
̺0(z) and length 2L, that is embedded in a uniform magnetic field
B = B0ẑ. Small amplitude torsional Alfvén waves in a cylindri-
cal coordinate system (r, φ, z), in which plasma profiles depend
on a longitudinal coordinate z only, can be described by the fol-
lowing linear equations:
4π̺0(z)
, (1)
, (2)
where uφ and bφ are the velocity and magnetic field components
of Alfvén waves.
These equations can be easily cast into a single wave equa-
V2A(z)
= 0 , (3)
where VA(z) = B0/
4π̺0(z) is the Alfvén speed. Assuming that
uφ ∼ exp(iωt), where ω is a wave frequency, we get the equation
uφ = 0 . (4)
For a trapped solution uφ must satisfy line-tying boundary con-
ditions which are implemented by setting
uφ(z = ±L) = 0 . (5)
Equation (4) with condition (5) consists the well-known Sturm-
Liuville problem which solution depends on the profile of VA(z).
We model the coronal loop by a rarefied plasma at the loop apex
(at z = 0) and by a compressed plasma at the loop footpoints
(z = ±L). Specifically, we adopt
̺0(z) = ̺00
1 + α2
, (6)
where ̺00 is the mass density at the loop apex and α
2 is a param-
eter which defines a strength of the inhomogeneity. For α2 = 0
the above mass density profile corresponds to a homogeneous
loop, while for a larger value of α2 the medium is more inhomo-
geneous. Figure 1 illustrates ̺0(z) for α
2 = 50. The mass density
is described by Eq. (6) with ̺00 = 10
−12 kg m−3 and L = 25 Mm.
Note that plasma is compressed at z = ±L. Substituting Eq. (6)
into Eq. (4), we obtain
1 + α2
uφ = 0 , (7)
where VA0 = B0/
4π̺00. With a use of the notation
y ≡ uφ, x ≡
z, a ≡ −
Eq. (7) can be rewritten in the form of Weber (parabolic cylinder)
equation (Abramowitz & Stegun 1964)
y = 0 . (9)
Fig. 1. Spatial profile of the background mass density, ̺0(z), given by
Eq. (6) with α2 = 50. The mass density and length are expressed in
units of 10−12 kg m−3 and 1 Mm, respectively.
Standard solutions to this equation are called Weber (parabolic
cylinder) functions (Abramowitz & Stegun 1964)
W(a,±x) =
(coshπa)1/4
G1y1(x) ∓
2G3y2(x)
, (10)
where
, G3 =
and y1(x), y2(x) are respectively even and odd solutions to
Eq. (9)
y1(x) = 1 + a
+ · · · ,
y2(x) = x + a
+ · · · .
2.1. Two limiting solutions
Periodic solutions to Eq. (9) can be written analytically in the
limiting cases: (a) for a large value of a but a moderate value
of x; (b) for a large x but a moderate a. The first (second) case
corresponds to α2 ≪ 1 (α2 ≫ 1).
2.1.1. Weakly inhomogeneous plasma
We consider first the case of a weakly inhomogeneous mass den-
sity field, i.e. α2 ≪ 1. In this case we have
a < 0, −a≫ x2, p ≡
−a . (12)
We adopt the following expansion (Abramowitz & Stegun
1964):
W(a, x) + iW(a,−x) =
2W(a, 0) exp [vr + i(px + π/4 + vi)] , (13)
where
W(a, 0) =
, (14)
vr = −
(x/2)2
(2p)2
2(x/2)4
(2p)4
+ · · · , vi =
2/3(x/2)3
+ · · · . (15)
T. Zaqarashvili & K. Murawski: Torsional oscillations of a coronal loop 3
As a result of relation −a≫ x2 we have from Eq. (13)
W(a, x) =
2W(a, 0) exp
cos ζ , (16)
W(a,−x) =
2W(a, 0) exp
sin ζ , (17)
ζ ≡ px + π/4 +
. (18)
The general solution to Eq. (9) is
uφ = c1W(a, x) + c2W(a,−x) , (19)
where c1 and c2 are constants.
For a homogeneous loop, i.e. α2 = 0, we recognize the well
known solution
uφ ∼ c1 cos (kz + π/4) + c2 sin (kz + π/4). (20)
Here wave number k satisfies the following homogeneous dis-
persion relation:
. (21)
Line-tying boundary conditions of Eq. (5) lead then to discrete
values of the wave frequency, viz.
1 + α2/6
, n = 1, 2, 3, . . . . (22)
From this dispersion relation we infer that in a comparison to
the loop with a homogeneous mass density distribution, ̺00, the
weakly inhomogeneous mass density field results in a decrease
of a wave frequency. This reduction is a consequence of the fact
that the inhomogeneous loop is denser at its footpoints, so the
average Alfvén speed is decreased. To show this, we compare the
results for the inhomogeneous loop with the homogeneous loop
with the same average density, so that both loops contain exactly
the same mass (Andries et al. 2005). We introduce a frequency
difference
∆ωn = ωn − ω̄n , (23)
where
ω̄n =
V̄A0 =
4π ¯̺0
corresponds to the average mass density
¯̺0 =
̺0(z)dz = ̺00
. (25)
Substituting Eq. (25) into Eq. (24), we obtain
ω̄n =
1 + α2/3
. (26)
From Eqs. (23) and (26) we find that ∆ωn ≤ 0. Here we infer
that in comparison to the average mass density case the wave
frequency is reduced, but as a result of α2 ≪ 1 the frequency
reduction is small. This is in a disagreement with Fermat’s law
and with the results of Murawski et al. (2004) who showed that
sound waves experience frequency increase in a case of a space-
dependent random mass density field.
2.1.2. Strongly inhomogeneous plasma
We discuss now a strongly inhomogeneous mass density case,
i.e. α2 ≫ 1. This case corresponds to x ≫ |a|. In this limit we
get (Abramowitz & Stegun 1964)
W(a, x) =
2k/x(s1(a, x) cos(ξ) − s2(a, x) sin(ξ)) , (27)
W(a,−x) =
2/kx(s1(a, x) sin(ξ) − s2(a, x) cos(ξ)) , (28)
where
− a ln x +
argΓ(1/2 + ia)
, (29)
1 + e2πa − eπa, (30)
s1(a, x) ∼ 1 +
1!2x2
2!22x4
− · · · , (31)
s2(a, x) ∼ −
1!2x2
2!22x4
+ · · · (32)
ur + ivr = Γ(r + 1/2 + ia)/Γ(1/2 + ia), r = 2, 4, . . . . (33)
The boundary conditions of Eq. (5) lead to the discrete fre-
quency spectrum
. (34)
Here we infer that the strongly inhomogeneous mass den-
sity field results in a significant decrease of a wave frequency in
comparison to the case of the loop with the constant density, ̺00.
This wave frequency decrease is a consequence of the fact that
the inhomogeneous loop is denser at its footpoints. Substituting
Eq. (34) into Eq. (23) we find that ∆ωn > 0. This wave frequency
decrease, in a comparison to the case of an average mass density
is now in an agreement with Fermat’s law and with the results of
Murawski et al. (2004).
3. Numerical results
Numerical simulations are performed for Eqs. (1), (2) with an
adaptation of CLAWPACK which is a software package de-
signed to compute numerical solutions to hyperbolic partial dif-
ferential equations using a wave propagation approach (LeVeque
2002). The simulation region (−L, L) is covered by an uniform
grid of 600 numerical cells. We verified by convergence studies
that this grid does not introduce much numerical diffusion and as
a result it represents well the simulation region. We set reflect-
ing boundary conditions at the left and right boundaries of the
simulation region.
Figure 2 shows a spatial profile of velocity uφ(z) for α
2 = 50,
drawn at t = 1000 s (solid line). This spatial profile results from
the initial Gaussian pulse that was launched at t = 0 in the center
of the simulation region, at z = 0. It is noteworthy that the sine-
wave profile of Eq. (20), which is valid for α2 = 0 (dashed line),
is distorted by the strong inhomogeneity which takes place for
the case of α2 = 50.
As a consequence of the inhomogeneity wave period is al-
tered. Figure 3 displays wave period P vs. inhomogeneity param-
eter α2. Diamonds represent the numerical solutions while the
solid lines correspond to the analytical solution to Eqs. (22) (top
panel) and (34) (bottom panel). Wave periods were obtained by
Fourier analysis of the wave signals that were collected in time
at the fixed spatial location, z = 0. It is discernible that the nu-
merical data fits quite well to the analytical curves. A growth of
4 T. Zaqarashvili & K. Murawski: Torsional oscillations of a coronal loop
Fig. 2. Numerically evaluated velocity profile uφ at t = 1000 s for α2 =
50 (solid line). This profile corresponds to the mode number n = 1. Note
that as a result of strong inhomogeneity, uφ departs from the sine-wave
which corresponds to α2 = 0. The dashed line corresponds to Eq. (20)
with c1 = c2 = 0.5.
Fig. 3. Wave period P = ω/2π vs. α2 for the mode number n = 1.
Diamonds correspond to the numerical solutions to Eqs. (1), (2). Solid
lines are drawn with the use of the analytical solution to Eqs. (22) and
(34). The wave period is expressed in seconds.
wave period P with α2 results from wave scattering on centers
of the inhomogeneity and it can be explained on simple physical
grounds. In an inhomogeneous field wave frequency ωn of the
torsional oscillations can be estimated from the following for-
mula:
V̄A0 , (35)
where V̄A0 is the averaged Alfvén speed that is expressed by Eq.
(24). Using P = 2π/ωn we obtain
4π ¯̺0
. (36)
As ¯̺0 grows with α, the growth of P with α results in.
4. Potential observations of torsional oscillations
Torsional oscillations of a coronal loop may result in periodic
variations of spectral line non-thermal broadening (expressed by
a half line width, ∆λB, hereafter HW) (Zaqarashvili 2003). For a
homogeneous loop, HW can be expressed as
∆λB =
uVA0λ
|sin(ωnt)sin(knz)| , (37)
where u is an amplitude of oscillations, λ is a wave length of
the spectral line and c is the light speed. Periodic variations of
spectral line width depend on a height above the solar surface:
a strongest variation corresponds to the wave antinode and the
place of a lack of line width variation corresponds to the nodes
(loop footpoints). Therefore, time series of spectroscopic ob-
servations may allow to determine a wave period. Knowing a
length of the loop, we may estimate the Alfvén speed, which in
turn gives a possibility to infer the magnetic field strength in the
corona. We estimate the expected value of line width variations
which result from torsional oscillations. For a typical coronal
Alfvén speed of ∼ 800 km/s, an amplitude of linear torsional
oscillation can be ∼ 40 km/s, which consists 5% of the Alfvén
speed. For the ”green” coronal line Fe XIV (5303 Å) from Eq.
(37) we obtain
∆λB ≈ 0.7 Å. (38)
This value is about twice larger than the original thermal broad-
ening of Fe XIV line. As a consequence, torsional oscillations
can be detected in time series of the green coronal line spectra.
For a weakly inhomogeneous distribution of mass density
along a loop, Eq. (22) enables to estimate the Alfvén speed at
the loop apex with the help of the observed period of HW vari-
ation and a loop length. For a strongly inhomogeneous density
profile along a loop, Eq. (34) shows that a wave period of tor-
sional oscillations is not just the ratio of the loop length to the
Alfvén speed, but it strongly depends on the rate of inhomo-
geneity, α2. Therefore, an additional effort is required in order
to apply the method of coronal seismology for torsional oscil-
lations. A spatial variation of mass density along the loop can
be estimated by a direct measurement of spectral line intensity
variation along the loop. Then, the estimated variation can be fit-
ted to Eq. (6), and hence a value of α2 can be inferred. Eq. (34)
provides a value of VA0 at the loop summit. Another possibility
is to collect time series of spectroscopic observations at differ-
ent positions of the loop. A spatial variation of line width along
the loop may be compared to the theoretical plot of uφ (Fig. 2),
which enables to estimate α2 and consequently Alfvén speed at
the loop apex (with a use of Eqs. (22) or (34)).
T. Zaqarashvili & K. Murawski: Torsional oscillations of a coronal loop 5
5. Discussion and summary
It is commonly believed that Alfvén waves are generated in the
solar interior either by convection (granulation, supergranula-
tion) or by any other kinds of plasma flow (differential rotation,
solar global oscillations). Due to their incompressible nature,
these waves may carry energy from the solar surface to the solar
corona and therefore they may significantly contribute to coro-
nal heating and solar wind acceleration. In closed magnetic loops
the Alfvén waves may set up the standing torsional oscillations,
while in opened magnetic structures these waves may propagate
up to the solar wind. As a result, observations of Alfvén waves
can be of vital importance to the problems of plasma heating and
particle acceleration.
The Alfvén waves that propagate along open magnetic field
lines may lead to a growth of a spectral line width with height
(Hassler et al. 1990, Banerjee et al. 1998; Doyle et al. 1998).
However, at some altitudes the spectral line width reveals a sud-
den fall off (Harrison et al. 2002; O’Shea et al. 2003, 2005). This
phenomenon was recently explained by resonant energy transfer
into acoustic waves (Zaqarashvili et al. 2006).
On the other hand, the photospheric motions may set up tor-
sional oscillations in closed magnetic loop systems, which can
be observed spectroscopically as periodic variations of spectral
line width (Zaqarashvili 2003). As a result, the observation of
Alfvén waves can be used as an additional powerful tool of coro-
nal seismology; the observed period and loop mean length en-
ables to estimate the Alfvén speed within a loop, which in turn
makes it possible to infer a mean magnetic field strength.
Besides their photospheric origin, torsional Alfvén waves
can be generated in the solar corona in a process of resonant ab-
sorption of the global oscillations (Ruderman & Roberts 2002,
Goossens et al. 2002, Andries et al. 2005, Terradas et al. 2006).
These oscillations may excite Alfvén waves in the outer inho-
mogeneous part of a loop, leading to attenuation of global oscil-
lations and amplification of torsional oscillations. These Alfvén
oscillations can be detected as periodic variations of spectral line
width. As a consequence, observations of Alfvén waves can be
a key for a determination of a damping mechanism of the loop
global oscillations.
Dynamics of torsional Alfvén waves in a homogeneous loop
can be easily solved. However, real coronal loops are longitudi-
nally inhomogeneous, which leads to alteration of wave dynam-
ics (Arregui et al. 2005, 2007, Van Doorsselaere et al. 2004a,b,
Donnelly et al. 2006, Dymova & Ruderman 2006, McEwan et
al. 2006). Therefore, the dynamics of Alfvén waves in longitudi-
nally inhomogeneous coronal loops must be understood in order
to provide analytical basis for potential observations of torsional
oscillations.
In this paper we discussed by analytical and numerical
means evolution of torsional Alfvén waves in an inhomogeneous
mass density field. The analytical efforts resulted in dispersion
relations which were obtained for a specific choice of an equilib-
rium mass density profile. These dispersion relations were writ-
ten explicitly for two limiting cases: (a) weekly inhomogeneous
and (b) strongly inhomogeneous mass density fields. From these
dispersion relations we inferred that the inhomogeneity results in
a wave frequency reduction in comparison to that of estimated at
the loop summit. This analytical finding is supported by the nu-
merical data which reveals that frequency reduction takes place
outside the region of validity of the analytical approach. As a
result of that we claim that a reduction of wave frequency is
ubiquitous for the inhomogeneous mass density field we consid-
ered. This reduction is a consequence of wave scattering on in-
homogeneity centers and it results from reduction of the average
Alfvén speed within a coronal loop. This frequency reduction
has important implications as far as wave observations are con-
cerned. The analytical formulae can be used for estimation of
coronal plasma parameters and therefore torsional Alfvén waves
consist an additional powerful tool of coronal seismology.
Acknowledgments: The authors express their thanks to the
referee, Prof. S. Poedts, for his stimulating comments. The work
of T.Z. is supported by the grant of Georgian National Science
Foundation GNSF/ST06/4-098. A part of this paper is sup-
ported by the ISSI International Programme ”Waves in the Solar
Corona”.
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	Introduction
	Analytical model of torsional oscillations
	Two limiting solutions
	Weakly inhomogeneous plasma
	Strongly inhomogeneous plasma
	Numerical results
	Potential observations of torsional oscillations
	Discussion and summary
