Global coronal seismology
arXiv:0704.1398v1  [astro-ph]  11 Apr 2007
GLOBAL CORONAL SEISMOLOGY
I. BALLAI
Solar Physics and Space Plasma Research Centre (SP2RC), Department of Applied
Mathematics, University of Sheffield, Hounsfield Road, Hicks Building, Sheffield,
S3 7RH, U.K. (i.ballai@sheffield.ac.uk)
Received ; accepted
Abstract.
Following the observation and analysis of large-scale coronal wave-like distur-
bances, we discuss the theoretical progress made in the field of global coronal seismol-
ogy. Using simple mathematical techniques we determine average values for magnetic
field together with a magnetic map of the quiet Sun. The interaction between global
coronal waves and coronal loops allows us to study loop oscillations in a much wider
context, i.e. we connect global and local coronal oscillations.
Keywords: Sun: magnetic field, Sun: waves, Sun: coronal seismology
1. Introduction
The possibility of using waves propagating in solar atmospheric plas-
mas to infer quantities impossible to measure (magnetic field, trans-
port coefficients, fine structuring, etc.) became a reality after high
cadence observation of oscillatory motion was made possible by space
and ground-based telescopes. These observations combined with the-
oretical models allow to develop a new branch of solar physics called
coronal seismology. Pioneering studies by Uchida (1970), Roberts, Ed-
win and Benz (1984), Aschwanden et al. (1999) and Nakariakov et al.
(1999), have formed the basis of a very promising and exciting field of
solar physics.
Traditionally, the terminology of coronal seismology was used mainly
to describe the techniques involving waves propagating in coronal loops.
Since then, this word has acquired a much broader significance and
the technique is generalised to acquire information about the mag-
netic solar atmosphere (De Pontieu, Erdélyi and James, 2004; Erdélyi,
2006). Coronal seismology uses waves which are localized to a par-
ticular magnetic structures, therefore it would be necessary to label
these seismic studies as local coronal seismology. After the discovery
of large-scale wave-like disturbances, such as EIT waves, X-ray waves,
etc., it became necessary to introduce a new terminology, i.e. global
coronal seismology where the information is provided by global waves
propagating over very large distances, sometimes comparable to the
solar radius. Although this may seem a separate subject, in reality these
c© 2018 Kluwer Academic Publishers. Printed in the Netherlands.
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http://arxiv.org/abs/0704.1398v1
2 BALLAI
two aspects of coronal seismology are very much linked. A global wave
generated by sudden energy releases (flares, CMEs) can interact with
active region loops or prominences and localized loop or prominence
waves and oscillations are emerging so, there must be a link between
the generating source and flare-induced waves in coronal loops.
Global waves have been known since the early 1960s. Although it is
still not known how the release of energy and energized particles will
transform into waves, today it is widely accepted that these distur-
bances are similar to the circularly expanding bubble-like shocks after
atomic bomb explosion or shock waves which follows the explosion of a
supernova. Thanks to the available observational facilities, global waves
were observed in a range of wavelengths in different layers of the solar
atmosphere. A pressure pulse can generate seismic waves in the solar
photosphere propagating with speeds of 200 – 300 kms−1 (Kosovichev
and Zharkova, 1998; Donea et al., 2006). Higher up, a flare generates
very fast super-alfvénic shock waves known as Moreton waves (Moreton
and Ramsey, 1960), best seen in the wings of Hα images, propagat-
ing with speeds of 1000 – 2000 kms−1. In the corona, a flare or CME
can generate an EIT wave (Thompson et al., 1999) first seen by the
SOHO/EIT instrument or an X-ray wave seen in SXT (Narukage et
al., 2002). There is still a vigourous debate how this variety of global
waves are connected (if they are, at all). Co-spatial and co-temporal
investigations of various global waves have been carried out but without
a final widely accepted result being reached. The present study deals
with the properties of EIT waves, therefore some characteristics of these
waves will be given below.
Unambiguous evidence for large-scale coronal impulses initiated dur-
ing the early stage of a flare and/or CME has been provided by the
Extreme-ultraviolet Imaging Telescope (EIT) observations onboard SOHO
and by TRACE/EUV. EIT waves propagate in the quiet Sun with
speeds of 250 – 400 km s−1 at an almost constant altitude. At a later
stage in their propagation EIT waves can be considered a freely propa-
gating wavefront which is observed to interact with coronal loops (see,
e.g. Wills-Davey and Thompson, 1999). Using TRACE/EUV 195 Å
observations, Ballai, Erdélyi and Pintér (2005) have shown that EIT
waves (seen in this wavelength) are waves with average periods of the
order of 400 seconds. Since at this height, the magnetic field can be
considered vertical, EIT waves were interpreted as fast MHD waves.
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Global coronal seismology 3
2. Coronal Global EIT Waves and their Applications
The observations of EIT waves propagating in the solar corona allowed
us to shed light on some elementary properties of coronal global EIT
waves, however, the available observational precision does not permit
us yet to determine more characteristics of these waves.
One of the un-answered problems related to EIT waves is connected
to their propagation. The core of the problem resides in the lack of
detection of an EIT wave with every flare or CME. This could be
explained partly by the poor temporal resolution of the SOHO satellite
(the only satellite giving full-disk EUV images at the moment) where
frames are available with a low cadence, therefore EIT waves generated
near the limb simply cannot be recorded. In general, EIT waves seen
by SOHO are generated by sources which are located near the centre
of the solar disk. EUV images provided by TRACE are much better
to use, although the field of view of this instrument is limited. Since
EIT waves propagate over a large area of the solar surface (at a certain
altitude) they are dispersive. Other ingredients to be considered are the
stratification of the medium and the inhomogeneous character of the
plasma. All of these factors influence the propagation of coronal EIT
waves.
Another plausible explanation for the absence of EIT waves asso-
ciated with every flare or CME might be that EIT waves diffuse very
rapidly, i.e. they become evanescent in a short time after their launch.
This means that only those EIT waves could be observed which propa-
gate as guided (trapped) waves. The MHD equations in a gravitationally-
stratified plasma allows as a solution the magnetoacoustic/magnetogravitational
waves of growing amplitude with time (upward propagating waves)
and decreasing amplitude with time (downward propagating waves).
Trapped EIT waves might arise as a combination of magnetoacoustic
and magnetogravitational waves propagating in opposite directions.
Further investigations of the possibility of trapping spherical waves in
a dissipative medium are needed.
2.1. Interaction of EIT waves with other coronal
magnetic entities
In this subsection we enumerate a few possible phenomena arising from
the interaction of global EIT waves with coronal magnetic entities.
According to the classical picture, EIT waves collide with coronal loops
resulting in a multitude of modes generated in loops either in the form
of standing oscillations or propagating waves. Both types of waves have
the general property that they decay very rapidly in a few wavelengths
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4 BALLAI
or periods (see e.g. Nakariakov et al., 1999; Aschwanden et al., 2002).
This damping was later used to diagnose the magnetic field inside
coronal loops (Nakariakov et al., 1999), transport coefficients for slow
waves or global fast waves, sub-structuring, heating function, etc.
In coronal loops we consider only the transversal generation of waves,
i.e. waves and oscillations are triggered by the interaction of EIT waves
and coronal loops. From the EIT wave point of view, a coronal loop
(similar to an active region or coronal hole) is an entity with a stronger
magnetic field (at least one order of magnitude) than the medium in
which they propagate (quite Sun). Therefore, beside transferring energy
to coronal loops, EIT waves can be scattered, reflected, and refracted
(Terradas and Ofman, 2004). Without claiming completeness, we can
draw a few conditions that could influence the appearance of coronal
loop oscillations:
- height of the loop: since EIT waves propagate at certain heights in the
solar corona, it is likely that not all loops will interact with the global
waves. Schrijver, Aschwanden and Title (2002) pointed out that only
those loops will be affected by EIT waves whose heights exceed 60 – 150
Mm. This means that cool, low-lying loops will not interact with EIT
waves.
- the height of the interaction between EIT waves and coronal loops:
this factor simply means that it is easier to generate oscillations in a
loop if the interaction point between the EIT wave and coronal loop is
closer to the apex of the loop rather than the footpoint.
- Orientation of the loop: if the front of the EIT wave is perpendicular
to the plane of the coronal loop the interaction between the EIT waves
and the coronal loop occurs in two points at the same time. If the loop
is stiff enough, a standing oscillation can be easily excited. If the front
is not perpendicular, the collision between the EIT wave and loops
occurs in two points delayed in time by τ = s cosα/vEIT , where s is
the distance between the footpoints, α is the attack angle, and vEIT is
the propagation speed of the EIT wave. In this case, standing modes
can be excited only in very special cases. Another important element
is the orientation of the coronal loop with respect to the vertical axis
(inclination).
- distance between the flaring site and coronal loop (or energy of EIT
waves): During their propagation, EIT waves are losing energy due
to the geometrical damping (dilatation of the front) and due to some
physical damping effects. Therefore it might happen that the energy of
an EIT wave originating from a distant flare is not enough to dislocate
the loop.
- radius of the loop and the density contrast (or Alfvén speed contrast):
a massive loop is much harder to dislocate than a thin loop. The ratio
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Global coronal seismology 5
between the densities in the loop and its environment is known to
influence the amplitude of oscillations.
In order to describe quantitatively the interaction between EIT
waves and coronal loops, we suppose a medium in which the coronal
loop is situated, for simplicity, in a magnetic-free medium (in fact this
constraint can be relaxed and the result is obvious) retains its identity
and does not disperse or fragment. The tube is considered thin, i.e.
its radius is small relative to other geometrical scales of the problem.
During the wave propagation we suppose a quasi-static pressure balance
to be maintained at all times.
An EIT wave colliding with a coronal loop exerts a force which will
need to work against two forces, one being the elastic force of the tube
represented by the magnetic tension of the tube and inertia of the fluid
element which needs to be displaced.
The equilibrium of the tube is prescribed by the hydrostatic equi-
librium where pressure forces are in equilibrium with the gravitational
force and the lateral pressure balance pi+B
i /2µ = pe is satisfied, with
pi and pe being the kinetic (thermal) pressure inside the tube and the
environment, Bi the interior magnetic field and g is the gravitational
acceleration at the solar surface. If we denote by ρi and ρe the locally ho-
mogeneous densities inside and outside the tube and vA(= Bi/(µρi)
the Alfvén speed, then the equation describing the variation of dis-
placement of the fluid element, ξ(z, t), is (a similar equation has been
obtained by Ryutov and Ryutova (1975) in a different context)
ρi − ρe
ρi + ρe
ρi + ρe
. (1)
Let us introduce a new variable such that ξ(z, t) = Q(z, t) exp(λz),
where the value of λ is chosen such that the first-order derivatives with
respect to the coordinate z vanish. After a straightforward calculation
we obtain that the dynamics of generated waves in the coronal loop as a
result of the interaction of a global wave with coronal loop is described
− c2K
+ ω2CQ = 0, ωC =
g(d− 1)
d(d+ 1)
cK = vA
1 + d
with d = ρi/ρe being the filling factor. Equation (2) is the well-known
Klein-Gordon (KG) equation derived and studied earlier in solar MHD
wave context by, e.g. Rae and Roberts (1982), Hargreaves (2005), Bal-
lai, Erdélyi and Hargreaves (2006). The quantity cK is the kink speed
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6 BALLAI
of waves and it is regarded as a density-weighted Alfvén speed. The co-
efficient ωC is the cut-off frequency of waves and is a constant quantity
for an isothermal medium.
The waves corresponding to the Eq. (2) are dispersive, i.e. waves
with smaller wavelength (larger k) propagating faster. Waves with
smaller wave number will have smaller group speed, the maximum of
the group speed (at k → ∞) being cK . Another essential property of
the KG equation is that it describes waves which are able to propagate
if their frequency is larger than the cut-off frequency. For typical coro-
nal conditions (vA=1000 km s
−1, d = 10) we obtain that waves will
propagate if their frequency is greater than 0.11 mHz or their period is
smaller than 150 minutes.
For simplicity, let us suppose that the fast kink mode in the coronal
loop is generated by the interaction of an EIT wave with a loop and
the forcing term of the interaction is modelled by a delta-pulse, i.e. the
equation describing the dynamics of impulsively generated fast kink
mode is given by
− c2K
+ ω2CQ = δ(z)δ(t). (3)
This equation can be solved using standard Laplace transform tech-
nique to yield
Q(z, t) =
2 − z2
H(cKt− |z|), (4)
where J0(z) is the zeroth-order Bessel function and H(z) is the Heav-
iside function. The impulsive excitation of waves in a flux tube leads
to the formation of a pulse that propagates away with the speed cK ,
followed by a wake in which the flux tube oscillates with the frequency
ωC . A typical temporal variation of the amplitude of kink waves (keep-
ing the height constant) would show that the amplitude of the mode
decreases (even in the absence of dissipation) and an e-fold decay occurs
in about 400 seconds .
Recently Terradas, Oliver and Ballester (2005) have studied the
interaction between the coronal loops and EIT waves in the zero-beta
limit considering a spatial initial condition. They obtained that the
generated oscillations in the coronal loop decay asymptotically as t−1/2.
Kink oscillations are weakly affected by dissipation, therefore the con-
sideration of any non-ideal effect to supplement the KG equation would
not lead to a significant change. It is accepted that damping due to the
resonant absorption could explain the damping of kink oscillations in
coronal loops (Ruderman and Roberts, 2002; Goossens, Andries and
Aschwanden, 2002).
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Global coronal seismology 7
It can be shown that the consideration of resonant absorption as a
damping mechanism in the governing equation leads to a similar equa-
tion we would obtain taking into account dissipation. Ballai, Erdélyi
and Hargreaves (2006) showed that the evolution equation is modified
by an extra term forming a Klein-Gordon-Burgers (KGB) equation
− c2K
+ ω2CQ− ν
∂z2∂t
= 0, (5)
where ν is a coefficient which could play the role of any dissipative
mechanism or a factor which include the damping due to resonant
absorption (in fact ν is inversely proportional to the gradient of Alfvén
speed) and would describe the transfer of energy from large to small
scales (see, e.g. Ruderman and Goossens, 1993).
Waves in this approximation can have a temporal (keeping k real)
and spatial damping (keeping ω real), the decay rate and length, sup-
posing the ansatz Q(z, t) ∼ exp[i(ωt− kz)], are given by
, ki ≈ −
ω2 − ω2C
c4K + ν
. (6)
The KGB equation can be solved using initial/boundary conditions to
describe the evolution of kink modes for different kind of sources, e.g.
monochromatic source (A(t) = V0e
iΩt), delta-function pulse (A(t) =
V0δ(ωCt/2π)), etc. using numerical methods. Asymptotic analysis (t ≫
z/cK) shows that these waves decay as t
−3/2 (Ballai, Erdélyi and Har-
greaves, 2006).
Another important factor is the energy of EIT waves. Recently
Ballai, Erdélyi and Pintér (2005), using a simple energy conservation,
found the minimum energy an EIT wave should have to produce a loop
oscillation. Using their results we studied a few loop oscillation events
presented by Aschwanden et al. (2002) and the minimum energy of
EIT waves necessary to produce the observed oscillations are shown in
Table 1. The geometrical size of loops and the number densities given
by Aschwanden et al. (2002) have been used.
The obtained energies are in the range of 1016 − 1019 J with no par-
ticular correlation with the length and radius of the loop. Similar to
this approach we can estimate the minimum energy of an EIT wave
to produce a displacement of 1 pixel in TRACE/EUV 195 Å images
using the relation E = 1.66 × 106L
2 + ρe/λ
, (J) where L and
R are the length and radius of the loop, and λ−1e the decay length of
perturbations outside the cylinder given by
λ2e =
(c2Se − c
Ae − c
(c2Se + v
Ae)(c
Te − c
k2, (7)
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8 BALLAI
Table I. The minimum energy of EIT waves which could produce the
loop oscillations studied by Aschwanden et al. (2002).
Date(yyyymmdd) L(Mm) R(Mm) n(×108 cm−3) E(J)
1998 Jul 14 168 7.2 5.7 2.2× 1017
1998 Jul 14 204 7.9 6.2 9.7× 1018
1998 Nov 23 190 16.8 3 1.3× 1019
1999 Jul 04 258 7 6.3 3.9× 1016
1999 Oct 25 166 6.3 7.2 1.6× 1018
2000 Mar 23 198 8.8 17 5.2× 1016
2000 Apr 12 78 6.8 6.9 2.5× 1016
2001 Mar 21 406 9.2 6.2 7.4× 1016
2001 Mar 22 260 6.2 3.2 1.9× 1016
2001 Apr 12 226 7 4.4 1.4× 1018
2001 Apr 15 256 8.5 5.1 1.4× 1016
2001 May 13 182 11.4 4 2.2× 1018
2001 May 15 192 6.9 2.7 1.6× 1019
2001 Jun 15 146 15.8 3.2 1.1× 1017
with cTe, cSe, and vAe being the cusp, sound and, Alfvén speeds in the
region outside the loop and k is the wavenumber. The energy range is
in the interval 3× 1017 − 3× 1018 J for loop lengths and radii varying
in the intervals 60 – 500 Mm and 1 – 10 Mm.
2.2. Determination of magnetic field values
Observations show that EIT waves propagate in every direction almost
isotropically on the solar disk, therefore we can reasonably suppose that
they are fast magnetoacoustic waves (FMWs) propagating in the quiet
Sun perpendicular to the vertical equilibrium magnetic field. The repre-
sentative intermediate line formation temperature corresponding to the
195 Å wavelength is 1.4×106 K. The sound speed corresponding to this
temperature is 179 km s−1. Since the FMWs propagate perpendicular
to the field, their phase speed is approximated by (c2S + v
The propagation height is an important parameter as a series of
physical quantities (density, temperature, etc.) in the solar atmosphere
have a height dependence. Given the present status of research on the
propagation of EIT waves, there is no accepted value for the propa-
gation height of these waves. For a range of the plasma parameters
we can derive average values for the magnetic field by considering
the propagational characters of EIT waves. Therefore, we study the
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Global coronal seismology 9
variation of various physical quantities with respect to the propagation
height of EIT waves.
We recall a simple atmospheric model developed by Sturrock, Wheat-
land and Acton (1996). The temperature profile above a region of the
quiet Sun, where the magnetic field is radial, is given by
T (x) =
7R⊙F0
)]2/7
. (8)
Here F0 is the inward heat flux (1.8× 10
5 erg cm−2 s−1), x is the nor-
malized height coordinate defined by x = r/R⊙, T0 is the temperature
at the base of the model (considered to be 1.3 × 106 K) and a is the
coefficient of thermal conductivity. The quantity a is weakly dependent
on pressure and atmospheric composition; for the solar corona a value
of 10−6 (in cgs units) is appropriate (Nowak and Ulmschneider, 1977).
Assuming a model atmosphere in hydrostatic equilibrium we obtain
that the number density, based on the temperature profile supposed in
Eq. (8), is
n(x) =
T (x)
exp[−δ(T (x)5/2 − T
0 )], δ =
2µGM⊙mpa
, (9)
with G the gravitational constant, M⊙ the solar mass, kB is the Boltz-
mann’s constant; µ = 0.6 is the mean molecular weight; mp, proton
mass and n0 = 3.6 × 10
8 cm−3 the density at the base of corona.
Having the variation of density with height and the value of Alfvén
speed deduced from the phase speed of EIT waves, we can calculate
the magnetic field using B = vA(4πmpn)
1/2. Evaluating the relations
above, the variation of temperature, density, Alfvén speed and magnetic
field with height is shown in Table 2. Two cases are derived for EIT
waves propagating strictly perpendicular to the radial magnetic field
with a speed of (a) 250 km s−1 and, (b) 400 km s−1, respectively.
The values of the physical quantities show some change for a given
propagation speed but will have little effect on the final results.
For an average value of EIT wave speed of 300 km s−1 propagating
at 0.05 R⊙ above the photosphere we find that the magnetic field is
1.8 G. If we apply Br2 = const., i.e. the magnetic flux is constant,
we find that at the photospheric level the average magnetic field is
2.1 G which agrees well with the observed solar mean magnetic field
(Chaplin et al. 2003). EIT waves considered as fast MHD waves can also
be used to determine the value of the radial component of the magnetic
field at every location allowed by the observational precision. In this
way, using the previously cited TRACE observations we can construct
a magnetic map of the quiet Sun (see Figure 1), in other words EIT
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10 BALLAI
Table II. The variation of the temperature (in MK), den-
sity (in units of 108 cm−3), Alfvén and sound speeds (in
units of 107 cm s−1) and magnetic field (in G) with height
above the photosphere for an EIT wave propagating with a
speed of (a) 250 km s−1, and (b) 400 km s−1, respectively.
r/R⊙ T n cS v
B(a) v
1.00 1.30 3.60 1.72 1.81 1.57 3.61 3.13
1.02 1.41 3.30 1.80 1.73 1.44 3.57 2.97
1.04 1.50 3.10 1.85 1.67 1.34 3.54 2.85
1.06 1.58 2.95 1.90 1.61 1.27 3.51 2.76
1.08 1.64 2.83 1.94 1.57 1.21 3.49 2.69
1.10 1.70 2.73 1.97 1.52 1.15 3.47 2.63
Figure 1. Magnetic map of the quite Sun obtained using an EIT wave observed by
TRACE/EUV in 195 Å .
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Global coronal seismology 11
waves can serve as probes in a magnetic tomography of the quiet Sun.
If points are joined across the lines, we will obtain the location of the
EIT wavefront. Magnetic field varies between 0.47 and 5.62 G, however,
these particular values should be handled with care as the interpolation
will introduce spurious values at the two ends of the interval. It should
be noted that this result has been obtained supposing a single value
for density, in reality both magnetic field and density can vary along
the propagation direction, as well. The method we employed to find
this magnetic map (magnetic field derived via the Alfvén speed) means
that density and magnetic field cannot be determined at the same time.
Further EUV density sensitive diagnostics line ratio measurements are
required to establish a density map of the quiet Sun which will provide
an accurate determination of the local magnetic field.
In conclusion, EIT waves propagating in the solar corona exhibit
a wide range of applicabilities for plasma and field diagnostics. The
fact that during their propagation EIT waves cover a large area of the
solar surface (in the coronal) allows us to sample the magnetic field
in the quiet Sun. EIT waves could serve as a link between eruptive
events and localised oscillations, e.g. loop oscillations could be studied
in a much broader context. Using a simple model we found that the
minimum energy an EIT wave should have to produce a detectable loop
oscillation is in the range of 1016 − 1019 J.
Problems to be tackled in the future should include the study of at-
tenuation of EIT waves with the aim of providing information about the
magnitude of transport coefficients in the quiet Sun. The old problem
of connecting different global waves still remain to be addressed.
Despite of the lack of high precision observations, EIT waves show
a great potential for magneto-seismology of the solar corona.
Acknowledgements
The author acknowledges the financial support offered by the Nuffield
Foundation (NUF-NAL 04) and NFS Hungary (OTKA, T043741). The
help by B. Pintér and M. Douglas is appreciated.
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