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Spatiotemporal Analysis of Coronal Loops Using Seismology of Damped Kink
Oscillations and Forward Modeling of EUV Intensity Profiles
D. J. Pascoe
1,2
,S.A.Anfinogentov
1,3
, C. R. Goddard
1
, and V. M. Nakariakov
1,4
1
Centre for Fusion, Space and Astrophysics, Department of Physics, University of Warwick, CV4 7AL, UK; david.pascoe@kuleuven.be
2
Centre for Mathematical Plasma Astrophysics, Mathematics Department, KU Leuven, Celestijnenlaan 200B bus 2400, B-3001 Leuven, Belgium
3
Institute of Solar-Terrestrial Physics SB RAS, Lermontov St. 126, Irkutsk 664033, Russia
4
School of Space Research, Kyung Hee University, Yongin, 446-701, Gyeonggi, Republic of Korea
Received 2017 December 20; revised 2018 May 2; accepted 2018 May 2; published 2018 June 8
Abstract
The shape of the damping profile of kink oscillations in coronal loops has recently allowed the transverse density
profile of the loop to be estimated. This requires accurate measurement of the damping profile that can distinguish
the Gaussian and exponential damping regimes, otherwise there are more unknowns than observables. Forward
modeling of the transverse intensity profile may also be used to estimate the width of the inhomogeneous layer of a
loop, providing an independent estimate of one of these unknowns. We analyze an oscillating loop for which the
seismological determination of the transverse structure is inconclusive except when supplemented by additional
spatial information from the transverse intensity profile. Our temporal analysis describes the motion of a coronal
loop as a kink oscillation damped by resonant absorption, and our spatial analysis is based on forward modeling the
transverse EUV intensity profile of the loop under the isothermal and optically thin approximations. We use
Bayesian analysis and Markov chain Monte Carlo sampling to apply our spatial and temporal models both
individually and simultaneously to our data and compare the results with numerical simulations. Combining the
two methods allows both the inhomogeneous layer width and density contrast to be calculated, which is not
possible for the same data when each method is applied individually. We demonstrate that the assumption of an
exponential damping profile leads to a significantly larger error in the inferred density contrast ratio compared with
a Gaussian damping profile.
Key words: magnetohydrodynamics (MHD) – Sun: corona – Sun: magnetic fields – Sun: oscillations –
Sun: UV radiation – waves
1. Introduction
Standing kink oscillations were first detected in coronal loops
using the Transition Region And Coronal Explorer (TRACE;
Aschwanden et al. 1999, 2002; Nakariakov et al. 1999). Their
detection is now routine (e.g., Zimovets & Nakariakov 2015;
Goddard et al. 2016), and thanks to the increased spatial and
temporal resolution of modern instruments such as the Atmo-
spheric Imaging Assembly (AIA; Lemen et al. 2012) of the
Solar Dynamics Observatory (SDO), they have great potential
for seismological investigation of the coronal plasma (e.g.,
reviews by De Moortel & Nakariakov 2012; Stepanov et al.
2012; Pascoe 2014; De Moortel et al. 2016). They are
commonly used to infer the strength of the coronal magnetic
field
(e.g., Nakariakov et al. 1999; Nakariakov & Ofman 2001;
Van Doorsselaere et al. 2008; White & Verwichte 2012; Pascoe
et al. 2016b; Sarkar et al. 2016). Additional structuring
information may be obtained using higher harmonics, which
there is increasing evidence of (e.g., Verwichte et al. 2004;De
Moortel & Brady 2007; Van Doorsselaere et al. 2007; Wang
et al. 2008; Srivastava et al. 2013; Pascoe et al. 2016a, 2017a,
2017c; Li et al. 2017). The strong damping of kink oscillations
is attributed to resonant absorption (Sedláček 1971), which
requires a smooth transition between the high-density plasma
inside coronal loops and the background plasma. Inside this
inhomogeneous layer, energy is transferred from kink to Alfvén
waves where the local Alfvén speed matches the kink speed C
k
on a timescale comparable to the period of oscillation (e.g.,
Hollweg & Yang 1988; Goossens et al. 2002; Ruderman &
Roberts 2002).
The damping rate due to resonant absorption depends on the
transverse density profile of the coronal loop. The observed
damping rate can therefore be used in seismological analysis of
oscillations to obtain information about the transverse structur-
ing. However, the damping rate is a single observable, whereas
models for the transverse density profile are typically described
by two unknowns (e.g., density contrast ratio and the width of
the inhomogeneous layer ). Furthermore, different models for
the shape of the density profile have been considered (e.g., a
sinusoidal or linear density profile inside the inhomogeneous
layer), and this choice also affects the expected damping rate of
kink oscillations (e.g., Goossens et al. 2002; Roberts 2008;
Soler et al. 2014).
Pascoe et al. (2013) proposed a seismological method based
on observing the initial Gaussian damping regime of kink
oscillations in addition to the later exponential damping regime.
The general damping profile (GDP ) that describes these two
regimes allows two observables to be measured, and hence the
transverse density profile can be calculated (for the assumed
density profile model). Pascoe et al. (2016b) analyzed the
transverse oscillations of three coronal loops using the GDP for
the first time to produce seismological inversions for the
transverse density profile. This method was extended by Pascoe
et al. (2017a) to describe a time-dependent period of
oscillation, the presence of additional parallel harmonics, and
any decayless component. Pascoe et al. (2017a) also employed
a new method to account for a dynamical background trend and
used Bayesian analysis to test the model against the
observational data. Pascoe et al. (2017c) included an additional
term in the background trend to describe loops that experience
The Astrophysical Journal, 860:31 (17pp), 2018 June 10 https://doi.org/10.3847/1538-4357/aac2bc
© 2018. The American Astronomical Society. All rights reserved.
1
a rapid shift in the location of the equilibrium, such as the
contracting loops previously analyzed by Simões et al. (2013)
and Russell et al. (2015). These applications of the seismolo-
gical method based on the shape of the damping profile exploit
the best cases of kink oscillations that have been observed so
far. However, the usefulness of seismological methods is
increased by extending the number of observations to which
they may be applied. In cases where the particular observa-
tional conditions are poor and less information can be reliably
extracted from the oscillation data, it is necessary to fill the gap
with some other source of information.
Another such source of information about the structure of
coronal loops that we consider in this paper is their intensity
profile. EUV imaging instruments also allow us to investigate
the transverse damping profile using the observed intensity
profile, for example, initial studies by Aschwanden et al.
(2003, 2007) and Aschwanden & Nightingale (2005) using
TRACE 171 Å. Pascoe et al. (2017b) used a similar method
(applied to data from SDO/AIA) to estimate the inhomoge-
neous layer width of a coronal loop and found a value
consistent with that calculated seismologically for the same
loop in Pascoe et al. (2017a). Goddard et al. (2017) studied
the density profiles of 233 coronal loops and found that the
majority exhibit evidence for having an inhomogeneous layer
of finite width. The potential for seismological techniques to
provide structuring information is also relevant for studies of
multi-threaded coronal structures considered by many authors
(e.g., Lenz et al. 1999; Aschwanden et al. 2000; Pascoe
et al. 2007; Brooks et al. 2012; Antolin et al. 2015;
Aschwanden & Peter 2017).
In this paper, we perform a detailed analysis of a particular
coronal loop, described in Section 2. The temporal (seismolo-
gical) and spatial
(forward modeling) diagnostic methods used
are described in Section 3. In Section 4, we apply each of these
methods separately to the loop data. In response to the strengths
and limitations of the single-model approach, we develop
techniques based on using multiple sources of data simulta-
neously, and the results of these multi-model methods are
presented in Section 5. Further discussion of effects relevant to
our method and results are in Section 6, including a comparison
with numerical simulations to estimate the associated errors.
Conclusions are presented in Section 7.
2. Observation
The coronal loop we analyze in this paper is shown in
Figure 1. It is designated as “Event 5 Loop 1” in Goddard
et al. (2016) and Pascoe et al. (2016c). In this paper, we use
the seismological methods for damped kink oscillations
appliedinPascoeetal.(2016b, 2017a) and the method of
forward modeling the EUV intensity profile from Pascoe
et al. (20 17b). We apply these methods to estimate model
parameters, in particular ò=l/R, which is the size of the
transverse inhomo geneou s layer l normalized by the loop
minor radius R.Forthesemethods,werequirethetimeseries
for the position of the loop and the intensity profile
perpendicular to the loop axis. In our case, both of these
observations are generat ed from t he same slit (blue line in
Figure 1) using the same bandpa ss (171 Å) of the sam e
instrument (SDO/ AIA ), though this is not a requirement. A
benefit of the multi-model a nalysis (Section 5) is i ts potenti al
to combine data from different sources.
The bottom panel of Figure 1 shows the time–distance map
for the loop starting at 04:40 on 2011 February 10, just after a
B6.0 GOES-class flare that occurs at 04:39 (Zimovets &
Nakariakov 2015). The white line shows the position of the
loop previously used in Pascoe et al. (2016c), which was based
on the assumption of a monolithic (i.e., single-thread) structure
and tracked in time using the standard method of fitting each
intensity profile using a Gaussian function (e.g.,
GAUSSFIT in
IDL) and determining the loop position as the center of the
Gaussian. However, on closer inspection, and by considering
the orientation of the loop (top panel of Figure 1), we note that
the line of sight (LOS) is almost parallel to the loop’s plane,
and hence the intensity profile is composed of the two legs
appearing to overlap on the plane of the sky. To accurately
measure the oscillation, we performed a fit using a model
comprised of two structures to allow us to track each leg of the
loop separately (green and blue lines in Figure 1). We note that
the legs oscillate in phase with each other and with the original
time series (white line), consistent with a horizontally polarized
fundamental kink mode that we are viewing almost side-on.
Fitting two legs rather than a single structure also produces a
lower and more accurate estimate of the loop minor radius
R≈7 Mm, though this is still relatively wide for a coronal
loop, which facilitates the estimation of the loop density profile
by forward modeling. The length of the loop is estimated to be
L=440±44 Mm. Accordingly, R/L≈0.02, consistent with
the thin tube approximation R/L=1.
3. Methods
In this section, we describe two methods that we apply to our
observational data individually in Section 4 and simultaneously
in Section 5.
3.1. Seismology Using Damped Kink Oscillations
The seismological method is based on the use of the kink
oscillation damping profile as described in Pascoe et al.
(2016b, 2017a, 2017c). Large-amplitude standing kink oscilla-
tions are typically observed for fewer than six cycles (e.g.,
Figure 2 of Goddard & Nakariakov 2016). Since our method is
based on the measurement of the damping profile, we require
this strong damping to be present in the signal. However,
compared to previous applications of the method, the data for
the oscillating loop analyzed in this paper are poor in terms of
having a higher noise level and hence a lower signal quality
(fewer cycles observed above the level of the noise).We
therefore consider a simpler model than in our previous papers,
based on the fundamental standing mode with amplitude A
1
and period of oscillation P
1
without the additional parallel
harmonics and initial shift in equilibrium position used in
Pascoe et al. (2017c),
yt A t
t
Pt
ysin
2
,1
1
1
trend
p
=+
⎛
⎝
⎜
⎞
⎠
⎟
() (
˜
)
˜
(
˜
)
()
where t
0
is the start time of the oscillation and
ttt
0
=-
˜
. The
period of oscillation
Pt
1
(
˜
)
is allowed to vary linearly with time
(compared with the third-order polynomial used in Pascoe
et al. 2017a). The background trend y
trend
is described by a
spline using six interpolation points per leg. Here
n
is the
2
The Astrophysical Journal, 860:31 (17pp), 2018 June 10 Pascoe et al.
damping envelope given by the GDP
t
t
tt
A
tt
tt
P
P
t
P
exp
2
exp
,
2
,
4
,
,2
2
g
2
s
s
s
d
s
g
1
12
d
1
2
s
1
t
t
t
pk
t
pk
k
=
-
-
-
>
=
=
=
⎧
⎨
⎪
⎪
⎩
⎪
⎪
⎛
⎝
⎜
⎜
⎞
⎠
⎟
⎟
⎛
⎝
⎜
⎞
⎠
⎟
()
()
where
A
tt
nss
==()
and
0e0e
krrrr=- +()()
. The
transverse density profile is described by the density contrast
ratio ρ
0
/ρ
e
and the width of the inhomogeneous layer ò. Here,
τ
d
is the exponential damping time ( e.g., Goossens
et al. 1992, 2002), and the corresponding time for the Gaussian
regime τ
g
is from Hood et al. (2013) using the thin tube
approximation C
k
=λ/P and the change in variable t=z/C
k
(see also Section6.1 of Pascoe et al. 2013). The applicability of
the approximation we use has also been demonstrated in
numerical simulations by Magyar & Van Doorsselaere (2016),
showing agreement within 10% for low-amplitude oscillations
approximating the linear regime. Since additional parallel
harmonics are not considered, neither are longitudinal structur-
ing effects, e.g., due to stratification or expansion. The damping
profile is based on the thin tube approximation, and so we also
neglect geometrical dispersion, that is,
PP LC2
1k k
==
,
where L is the loop length, and the kink speed for a low-β
plasma (uniform magnetic field) is
CC
2
1
,3
kA0
e0
rr
=
+
()
where C
A0
is the internal Alfvén speed, and the external Alfvén
speed is given by
C
C
Ae A0
0e
rr=
. When this model is
applied to the observational data, we instead consider the
period of oscillation in terms of the Alfvén transit time inside
the flux tube T
A
=L/C
A0
, giving
PT2
1
2
.4
kA
e0
rr
=
+
()
This has the benefits of separating out the dependence on the
density contrast, which is already a model parameter because it
also affects the damping profile, and not requiring the loop
length to be included in the model, since that parameter is
estimated independent of our seismological method—for
example, by magnetic extrapolation (e.g., Verwichte et al.
2013; Long et al. 2017; Pascoe et al. 2017c) or, more simply, as
π times the height of the loop (accounting for the inclination of
the loop’s plane from the vertical). Since the period of
oscillation is allowed to vary linearly, it is described by two
values of the Alfvén transit time: T
A0
at time t
0
and T
A1
at the
end of the time series.
Phase mixing (e.g., Heyvaerts & Priest 1983) of the Alfvén
waves generated by mode coupling increases the efficiency of
dissipative processes. We can use the model parameters
discussed above to estimate the lifetime of the Alfvén waves
(Mann & Wright 1995; Pascoe et al. 2016b) as
L
CC
.5
A
Ae A0
t
p
=
-()
()
3.2. Forward Modeling of EUV Intensity Profile
In this section, we consider seven models for the transverse
density profile, described by Equations (6)–(12) below, and
apply each to our observational data, i.e., the transverse
intensity profile. Each profile describes an enhancement to the
transverse density profile due to a coronal loop with the
parameter A=ρ
0
−ρ
e
, where the internal density is greater
than the external density ρ
0
>ρ
e
, and ρ
e
>0. Since these
profiles describe the density enhancement, when comparing
the profiles to data, they are added on top of a background
density profile describing the plasma outside the loop, which is
Figure 1. SDO/AIA 171 Å image (top) of the analyzed loop with its axis
indicated by the dashed red line. The solid blue line shows the location of the
slit used to generate the time–distance map (bottom). The white line shows the
original time series used in Pascoe et al. (2016c) based on a single Gaussian fit
to the intensity profile, while the green and blue lines track the two loop legs
separately.
3
The Astrophysical Journal, 860:31 (17pp), 2018 June 10 Pascoe et al.
not necessarily uniform. For our data, the field of view is
closely cropped around the oscillating loop (Figure 1), and so
we find a linear background density profile to be sufficient to
account for its variation (Figure 5).
The transverse density profile used for our seismological
method (Section 3.1) is the linear transition layer profile
(Model L)
r
Arr
Arrr
rr
,
1,
0,
,6
rr
rr
1
12
2
1
21
r =
-<
>
-
-
⎧
⎨
⎪
⎪
⎩
⎪
⎪
()
()
∣∣
∣∣
∣∣
()
where r is the local coordinate across the flux tube at the point
of the observation, r
1
=R−l/2, r
2
=R+l/2, l= òR. Our use
of this profile is based on the availability of the full analytical
solution for the damping envelope (Hood et al. 2013), and so it
is currently the only profile that can be used for both our
temporal and spatial analysis. However, when considering the
spatial information revealed by the intensity profile separate
from the seismological method ( e.g., Goddard et al. 2017;
Pascoe et al. 2017b), we may choose any other profile to test.
Below, we describe six other density profiles we also test
against the observational data. The results of these tests
(Section 4.2) support our use of Model L, and this choice is
also discussed further in Section 6.2.
The step function profile (Model S) corresponds to the
limiting case ò=0 for which there is no smooth inhomoge-
neous layer, and so kink oscillations would not be subject to the
enhanced damping caused by resonant absorption (e.g., Edwin
& Roberts 1983):
r
Ar R
rR
,
0,
.7
r =
>
⎧
⎨
⎩
()
∣∣
∣∣
()
A Gaussian profile is commonly used to fit the transverse
intensity profile of the coronal loops. The transverse density
profile itself being Gaussian (Model G) has instead been
previously considered (e.g., Aschwanden et al. 2007; Pascoe
et al. 2017b):
rA
r
R
exp
2
.8
2
2
r =-
⎛
⎝
⎜
⎞
⎠
⎟
() ()
The generalized symmetric Epstein profile (Model E; e.g.,
Nakariakov & Roberts 1995; Pascoe & Nakariakov 2016) is
defined as
rA
r
R
sech , 9
p
2
r =
⎜⎟
⎛
⎝
⎞
⎠
()
∣∣
()
which describes a smooth profile, as with the Gaussian profile,
but with a controlled steepness determined by the parameter p.
Increasing the steepness corresponds to a more localized
inhomogeneous layer, and Model S is reproduced in the limit
p ¥
(although even values of p10 could be observa-
tionally indistinguishable).
The linear dependence used in Model L is not the only
possible choice for the density profile inside an inhomogeneous
layer. A sinusoidal (Model N) dependence has also been
considered by several authors (e.g., Ruderman & Roberts 2002;
Terradas et al. 2006):
r
Arr
rR
l
rrr
rr
,
1sin ,
0,
.10
A
1
2
12
2
r
p
=-
-
<
>
⎜⎟
⎧
⎨
⎪
⎪
⎩
⎪
⎪
⎛
⎝
⎞
⎠
()
∣∣
()
∣∣
∣∣
()
We note that Model N may also be used for seismological
analysis when only considering the exponential damping
regime (Sections 6.2 and 6.3).
Soler et al. (2013, 2014) also proposed an inhomogeneous
layer with a parabolic density profile (Model P) with the form
r
Arr
AArRl lr r r
rr
,
2,
0,
.11
1
22
12
2
r =
--+ <
>
⎧
⎨
⎪
⎩
⎪
()
∣∣
()∣∣
∣∣
()
Finally, a continuous profile based on the hyperbolic tangent
function (Model T) has also been used in numerical simulations
of oscillating coronal loops (e.g., Antolin et al. 2014; Howson
et al. 2017; Pagano & De Moortel 2017):
r
AerR
rr2
1tanh . 12
21
r =-
-
-
⎛
⎝
⎜
⎞
⎠
⎟
()
()
()
Our use of the isothermal approximation greatly simplifies
calculations by relating the intensity profile to the square of the
density integrated along the LOS, i.e., excluding the temper-
ature dependence of the EUV emission and the instrument
response function.
For Model S, the density inside the loop is constant, so the
integrated loop intensity per unit length along its axis is readily
found to be proportional to A
2
d, where d is the loop depth along
the LOS given by the chord length of the circular loop cross-
section
dRrrR2,, 13
22
=-∣∣ ( )
where
r
xx
0
=-∣
∣
, x is the direction transverse to the loop,
and the loop center is at x
0
. This method can be extended to the
other density profile models by considering the contributions
from a number of cylindrical shells, each having a uniform
density given by the corresponding density profile at the center
of the shell. For Models L, N, and P, the density in the core
region is constant, so the shells are uniformly distributed over
r
rr,
12
= [
]
. For Models G, E, and T, the density varies
continuously, so the shells are distributed over r=[0, 2.5R].
We use 100 shells in our analysis, which is sufficient to
produce converged results. The results are also consistent with
the approach based on a 2D array used in Pascoe et al. (2017b)
and Goddard et al. (2017), but the increased efficiency allows
us to readily investigate the variation of the transverse density
profile with time (Section 6.1).
Since the corona is optically thin, the contribution to the
intensity profile from the background plasma depends on the
LOS integration depth. This is generally unknown, and we
assume a value of 100Mm as a reasonable estimate. However,
for this reason, the EUV intensity profile cannot be used to
calculate the density contrast of the coronal loops directly.
Instead, we use forward modeling to estimate the spatial scale
of the transverse structuring (R and ò) and then combine this
with our seismological information to calculate the density
contrast ratio. Kink oscillations therefore allow us to probe the
4
The Astrophysical Journal, 860:31 (17pp), 2018 June 10 Pascoe et al.
