Characteristics of Transverse Oscillations in a Coronal
Loop Arcade
E. Verwichte1 , V.M. Nakariakov1 , L. Ofman2 and E.E. DeLuca3
1
2
University of Warwick, UK
Catholic University of America / NASA Goddard Space Flight Center, USA
3
Harvard-Smithsonian Center for Astrophysics,USA
Contact: Erwin.Verwichte@warwick.ac.uk
Preprint: http://www.astro.warwick.ac.uk/∼erwin/Publications/
Transverse oscillations in a coronal loop arcade – p.1/16
Introduction
Aschwanden et al. (1999) detect transverse loop oscillations with TRACE and
interpret them as standing fast magnetoacoustic kink waves, triggered by a
passing disturbance.
Comparison between such wave observations with MHD wave theory allows for
the determination of coronal physical parameters crucial to coronal heating models
(e.g. strength of magnetic field or dissipation).
Nakariakov et al. (1999) notice oscillation damping in the order of 10-15 min.
Various hypothesis are proposed to explain damping:
1. anomalously high shear viscosity or resistivity (Nakariakov et al., 1999)
2. photospheric loop footpoints topology (Schrijver & Brown, 2000)
3. phase-mixing through small-scale loop inhomogeneities (Roberts, 2000;
Ofman & Aschwanden, 2002)
4. energy transfer through resonant absorption (Ruderman & Roberts, 2002;
Goossens, Andries & Aschwanden, 2002)
Transverse oscillations in a coronal loop arcade – p.2/16
The observation sequence
TRACE 171 Å sequence of April 15th , 2001, between 22:00:43 and 22:27:50 UT,
with a resolution of 26 s temporally and 1 arcsecond spatially.
Target: AR NOAA 9415, then on the SW limb, which was active with a X14.4 flare
at 13:48 UT. A post-flare loop arcade was first seen by TRACE at 14:40 UT.
This arcade is disturbed by a prominence eruption nearby to the north, which
presumably is the cause for transverse oscillations in the loops of the arcade.
Subfield of an 195 Å
EIT/SoHO
full-disk
image at 20:24 UT
Subfield of TRACE
observation at 22:11
UT with paths used
in the analysis
Transverse oscillations in a coronal loop arcade – p.3/16
Description of oscillations
Because the active region is close to the solar limb and the arcade axis lies
quasi-meridionally, the spacecraft looks along the plane of the arcade loops. The
transverse loop oscillations appear as a back and forth motion of the loop plane.
The oscillation period is of the order of five minutes and 2-2.5 oscillation cycles
can be clearly distinguished. The oscillation amplitude decays with time.
Time-space diagram of path
following top of loop arcade
with North to the right.
.
Transverse oscillations in a coronal loop arcade – p.4/16
Loop superpositions
The scene is often confused due to the superposition along the line of sight of shifting
loops and/or the two branches of the same loop, causing loops in space-time diagrams
to appear double, confusing the oscillation amplitude or increasing the intensity.
=⇒ interactive selection of oscillation displacement.
The loop in path G appears at times double, with the two parts oscillating with the
same amplitude, period and phase. It is an example of superposition of two
branches of the same loop.
In path C two loops cross at the end of the sequence, giving the impression of a
growing oscillation amplitude (simulated below).
Simulation of the superposition of two crossing loops.
Transverse oscillations in a coronal loop arcade – p.5/16
Path parameters
Path
∆x
H
L
(Mm)
(Mm)
(Mm)
∆s/L
A
9
65
203
0.17
B
5
68
214
0.12
C
12
70
218
0.19
D
18
73
228
0.23
E
10
74
233
0.17
∆x is the analysed x-interval, H is the loop height
F
7
74
233
0.14
defined as the projected distance between the loop top
G
15
76
237
0.20
H
7
75
235
0.14
I
12
75
236
0.18
and the solar limb and L is the length of a circular loop
H . The analysed loop fraction, ∆s/L, is
the fraction of the circular loop that is covered by ∆x,
and is calculated as arccos((H − ∆x)/H)/π . If
with radius
the superposition of the loop legs is taken into account,
this fraction needs to be doubled.
Transverse oscillations in a coronal loop arcade – p.6/16
Oscillation measurement
A path is taken along a loop with width of 25 pixels. At each fixed point along the loop, a
space-time plot is contructed from a transverse slice and the oscillation displacement is
measured (e.g. path G).
Two methods:
1d wavelet transform with Morlet motherwavelet (Torrence & Compo, 1998).
curvefitting with function ξ(x, t) = A e−(t/τn )
estimates for A, P and φ from WT.
n
cos(2πt/P + φ) with starting
Transverse oscillations in a coronal loop arcade – p.7/16
Oscillation parameters
Path
A(x) (km)
453 ± 140 - (34 ± 27) x
A
B
φ(x) (deg)
(156 ± 188 + (30 ± 52) x)
342 ± 66 + ( 1 ± 12) x (a)
C
430 ± 109 - (25 ± 14) x (a)
D
(135 ± 66 + (14 ± 7)
x)(a)
487 ± 77 - (13 ± 9) x
424 ± 69 - (10 ± 12) x (c)
E
F
G
H
I
-
231 ± 38
184 ± 94
102 ± 112 (a)
7 ± 188 (a)
183 ±
9
certain measurements.
(a) Based upon estimates from
wavelet analysis.
(b) Linear trend in phase ob-
- (6.6 ± 1.7) x (b) served for x < 10 Mm.
(c) Based upon measurements
191 ± 18
for x < 7 Mm.
176 ± 143
557 ± 68 - (12 ± 14) x
171 ± 65
238 ± 86 - ( 6 ± 12) x
143 ± 76
(218 ± 50 - ( 8 ± 9) x) (a)
Values between brackets are un-
172 ± 92
Transverse oscillations in a coronal loop arcade – p.8/16
Oscillation parameters (bis)
Path
wavelet P
fit P
τ1
τ2
τ3
(s)
(s)
(s)
(s)
(s)
-
-
-
-
-
-
A
301 ± 50
B
-
C
247 ± 33
448 ± 18
D
242 ± 31
396 ± 20
326 ± 107
393 ± 77
315 ± 144
(980)(a)
920 ± 290
860 ± 300
(405 ± 35)
-
(1550 ± 640)
(1400 ± 320)
-
-
-
-
392 ± 31
F
-
G
346 ± 78
358 ± 30
1030 ± 680
1060 ± 420
920 ± 260
325 ± 107
357 ± 89
(960 ± 760)
1400 ± 580
1360 ± 650
I
317 ± 80
326 ± 45
-
1350 ± 480
1840 ±580
379 ± 54
243 ± 103
1320 ± 570
1780 ± 560
E
H
382 ± 12
1180 ± 1050
960 ± 420
1180 ± 340
-
1010 ± 380
930 ± 310
Values between brackets are uncertain measurements.
(a) Based upon one measurement.
Transverse oscillations in a coronal loop arcade – p.9/16
Dependency of displacement amplitude on distance
For a circular, homogeneous loop with its loop plane in the line-of-sight
fundamental standing mode:
(1)
A(x) = A(xtop ) − (A(xtop )/H)x ,
where H is the loop height. With A(xtop ) and H in the ranges 100-600 km and 65-76
Mm resp. ⇒ dA/dx ≈ -5 km Mm−1 < 0.
second harmonic standing mode:
(2)
A(x) = Amax sin(2 arccos((H − x + xtop )/H)) .
⇒ dA/dx > 0 near loop top.
√
Position of the maximum amplitude is xmax − xtop = (2 − 2)H/2.
For both loops C and D the value of H is in the range 70-73 Mm.
⇒ xmax = 20.5-21.4 Mm, ≈ 20 Mm from the loop top.
Transverse oscillations in a coronal loop arcade – p.10/16
Multiple oscillations
Path D
Top left: magnitude of |WT| at
x = 11 Mm
from looptop. Two periods are detected: 240
and 400 s.
Top right:
and :
P (x). •: WT. +: n=1, ∗: n=2
n=3 curvefitting.
Bottom left: A(x) for P 400 s. • is WT. Each
dataset is fitted by a straight line (−−: n=1,
−.−: n=2, − · · · −: n=3 and −−: WT).
Bottom right: A(x) for P 240 s. Only WT
available as •. The dashed line is a linear
fit. The solid curves represent the profile of
a second harmonic standing mode for several
amplitude values.
Periods and amplitude profile as a function of distance suggest that P 400 s and P 240 s
are the fundamental and second harmonic standing fast kink wave respectively.
Transverse oscillations in a coronal loop arcade – p.11/16
Deriving plasma quantities
The transverse oscillation is considered to be a standing fast magnetoacoustic
kink oscillation:
q
2L
2
VA with Ck the kink speed.
Vphase = nP ≈ Ck =
1+ρ /ρ
e
0
=⇒ Alfvén speed VA and magnetic field strength B. Assuming Ne = 1-6 1015
m−3 and ρ = 0-0.3 ρe , find ranges 600-1800 km s−1 and 9-46 G for nine loops
resp., consistent with previous results (Nakariakov & Ofman, 2001).
Two hypothesis: phase mixing predicts τ ∼ P 4/3 , ideal mode conversion (resonant
absorption) predicts τ ∼ P (` constant). Ofman & Aschwanden (2002) found
τ ∼ P 1.17±0.35 , which does not permitt to distinguish between the two
hypothesis.
phase-mixing is assumed =⇒ coronal, kinematic (bulk) viscosity coefficient ν.
Theory predicts a (compressive) Reynolds number R = LV A /ν ≈ 106 for nine
loops, with which the measurements are consistent.
if
Transverse oscillations in a coronal loop arcade – p.12/16
Deriving plasma quantities: results
Path
A
B
C
Ck
VA
B
ν
(km s−1 )
(km s−1 )
(G)
(m2 s−1 )
880 - 1000
13 - 36
-
-
770 - 880
11 - 31
-
-
620 - 700
9 - 25
690 - 790
10 - 28
0.6 - 1.0 × 108
670 - 760
10 - 27
-
-
820 - 940
12 - 33
0.3 - 0.4 × 108
860 - 980
13 - 35
0.8 - 1.5 × 108
4.4 - 7.6 × 106
1360 - 1550
20 - 55
-
-
940 - 1070
14 - 38
1.4 - 2.8 × 10 8
0.8 - 1.8 × 106
1020 - 1160
15 - 41
940 - 1070
14 - 38
0.6 - 0.9 × 10 8
2.6 - 4.1 × 106
1250 ± 410
1090 ± 210
880 ± 120(a)
970 ± 40
D
940 ± 120(a)
1160 ± 90
E
F
G
H
I
1220 ± 40
1920 ± 810
1320 ± 110
1440 ± 200
1320 ± 330
(a)
R
0.9 - 1.4 × 108
1.3 - 2.2 × 10 8
(a)
1.0 - 1.7 × 106
1.4 - 2.9 × 106
1.4 - 3.0 × 106
1.1 - 2.0 × 106
Oscillation is assumed to be a second harmonic.
Transverse oscillations in a coronal loop arcade – p.13/16
Damping time Vs. period
Slope of our measurements consistent with previous results.
The predictions of both hypothesis fall within one standard deviation from the
observational result ⇒ observations cannot distinguish.
Our measurements of the damping times have a bias towards longer times:
- post-flare loops are different than ordinary active region loops.
- nearby erupting prominence may be responsible for exciting oscillations over a certain
time period instead of a single excitation.
Transverse oscillations in a coronal loop arcade – p.14/16
Discussion
Analysis of multiple transverse oscillations in post-flare loop arcade in TRACE
sequence of April 15th , 2001, which are interpreted as fast magnetoacoustic
standing kink waves.
Oscillation characteristics are determined as a function of position along the loop.
Oscillation damping has similar period dependency as found earlier, but has a bias
towards longer times probably due to different excitation mechanism or loop
structure.
Consistent values for the coronal magnetic field strength and viscosity are derived.
For the first time, the second harmonic oscillation is observed.
Transverse oscillations in a coronal loop arcade – p.15/16
References
Aschwanden, M. J., Fletcher, L., Schrijver, C. J., Alexander, D.: 1999, Astrophys. J.
520, 880.
Aschwanden, M.J., De Pontieu, B., Schrijver, C.J., Title, A.M.: 2002, Solar Phys. 206,
99.
Goossens, M., Andries, J., Aschwanden, M.J.: 2002, Astron. Astrophys 394, 39.
Nakariakov, V.M., Ofman, L., DeLuca, E.E., Roberts, B., Davila, J.M., 1999, Science
285, 862.
Nakariakov, V.M., Ofman, L.: 2001, Astron. Astrophys. 372, 53.
Ofman, L., Aschwanden, M.J.: 2002, Astrophys. J. 576, 153.
Roberts, B.: 2000, Solar Phys. 193, 139.
Ruderman, M.S., Roberts, B.: 2002, Astrophys. J. 577, 475.
Schrijver, C.J., Brown, D.S.: 2000, Astrophys. J. 537, L69.
Torrence, C., Compo, G.P.: 1998, Bull. Amer. Meteor. Soc. 79, 61.
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