4.5 MHD oscillations in coronal loops

A review of the 3D geometry, motion, and hydrodynamic aspects of oscillating coronal loops is given in Aschwanden (2009cJump To The Next Citation Point), which describes two types of 3D reconstructions of kink-mode loop oscillations: (i) Deprojection of 2D loop tracings using the method of curvature radius maximization, and (ii) stereoscopic triangulation of loop tracings, both applied as a function of time. The time-dependent 3D reconstruction of oscillating loops, [x(t),y(t),z(t)], yields vital diagnostic on the standing (eigen-modes) and/or propagating MHD waves, and on the linear and/or circular polarization of the excited wave modes. We describe here only the stereoscopic results.

In a survey of flare and CME events observed with the SECCHI/EUVI telescope during the first two years of the mission, at least 10 events were identified that contain loop oscillations and/or propagating waves (Aschwanden et al., 2009bJump To The Next Citation Point), which we list in Table 2 (Movies and quicklook plots of these oscillation events are also available at External Linkhttp://secchi.lmsal.com/EUVI/). This list of events may not be complete, because oscillating loops and propagating waves were mostly identified by visual inspection of EUVI 171 Å movies (either by some periodic motion over at least two periods or by some global wave propagation), rather than by automated detection algorithms that scan the entire EUVI database. Inspecting the spacecraft separation angle in Table 2 we see that 9 events have been observed at separation angles of αsep ≈ 6 ∘–47 ∘ and are thus suitable for stereoscopic 3D reconstruction.

Table 2: Flare and CME events with loop oscillations and waves observed with STEREO/EUVI (Aschwanden, 2009cJump To The Next Citation Point).
# Date Time Heliogr. Stereo GOES
  location angle class E P A B Com. Rep.
  (UT)   (deg)   (keV) (cts/s)
(105 DN/s)
8 07/01/12 01:00 – 03:00 N00W87 0.2 C1.5 12 34 31.2 ... IPEW L
35 07/05/02 23:20 – 23:59 S09W16 6.3 C8.5 12 30 23.0 17.3 IPW ...
43 07/05/16 17:10 – 18:10 N03E34 8.2 C2.9 12 26 12.7 9.9 IPDW LS
46 07/05/19 12:40 – 13:20 N03W03 8.6 B1.3 12 46 17.4 13.9 IPEDW LS
100 07/06/08 12:30 – 13:30 S08W08 11.9 B7.6 12 28 7.9 5.9 IPDW LS
104 07/06/09 13:20 – 14:40 S10W23 12.1 M1.0 12 184 69.4 45.8 IW L
109 07/06/27 17:30 – 18:30 S20E89 15.4 C1.3 3 34 2.9 2.4 IDOW ...
133 07/08/06 15:20 – 15:50 S12E38 23.3 C1.1 12 32 7.8 25.2 IPW LS
175 07/12/31 00:30 – 01:40 S15E87 44.0 C8.3 25 288 7.5 39.3 IPDOW LS
183 08/03/25 18:30 – 19:30 S25E90 47.2 M1.7 12 944 17.8 192.2 IPDEW LS
#) Event identification number in EUVI flare and CME catalog:
External Linkhttp://secchi.lmsal.com/EUVI/euvi_events.txt
     E = highest detected energy range: 3 – 6, 6 – 12, 12 – 25, 25 – 50, 50 – 100 keV,
     P = peak count rate of RHESSI light curve.
2) EUVI comments:
     A = background-subtracted peak flux detected in EUVI/A,
     B = background-subtracted peak flux detected in EUVI/B,
     D = Dimming in EUV,
     E = Eruptive feature,
     I = Impulsive EUV emission (simultaneous with hard X-rays),
     O = Occulted (for A if flare position is east, or for B if west),
     P = Postflare loop emission,
     W = Waves or oscillations.
3) CME reports:
     L = LASCO/SOHO,
     S = SECCHI Cor-1 or Cor-2.

One loop oscillation event occurred during the 2007 Jun 27, 17:30 UT, flare and has been analyzed in several studies (Verwichte et al., 2009Jump To The Next Citation Point; Aschwanden et al., 2009bJump To The Next Citation Point; Aschwanden, 2009cJump To The Next Citation Point). No stereoscopic 3D reconstruction with a triangulation method has been attempted in the first study on this loop, because the loop footpoints are not visible in one spacecraft (STEREO/A) and cannot easily be located in the image of the other spacecraft (STEREO/B), as well as corresponding features of the upper loop segment are difficult to verify in both images (Verwichte et al., 2009Jump To The Next Citation Point). Instead, a semi-circular loop geometry was fitted in projection to both stereoscopic images by varying the inclination angle of the loop plane, leading to a loop radius of rloop = 110 Mm, an azimuth angle (of the loop baseline) of α = 29∘ (anti-clockwise from the heliographic east-west direction), and an inclination angle of ∘ 𝜃 = 27. The 2D projection of this 3D geometry in the LOS directions of STEREO/A and B is shown in Figure 31View Image. The time behavior of the oscillating loop is shown in a sequence of running-difference images in Figure 32View Image, which displays clear kink-mode displacements (black/white loop rims) for the considered loop. A time-slice plot obtained along a cross-sectional slice reveals the damped sinusoidal oscillation (Figure 33View Image), whose amplitude could be fitted by

( 2π(t − t0) ) ( t − t0 ) a(t) = a0 + a1 cos ----------+ Φ exp − ------ , (42 ) P τd
yielding an initial amplitude of a1 = 2.5 EUVI pixels (2900 km), a period of P = 565 s (≈ 9 min), and an exponential damping time of τd = 1600 s (≈ 27 min), similar to the values found by Verwichte et al. (2009Jump To The Next Citation Point). Note that the EUVI cadence is 150 s (2.5 min).
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Figure 31: 3D reconstruction of an oscillating loop observed during the flare of 2007 Jun 27, ≈ 18:19 UT, superimposed on the STEREO/A (left) and STEREO/B images (right), obtained by stereoscopic fitting of a circular loop geometry (from Verwichte et al., 2009Jump To The Next Citation Point).
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Figure 32: Sequence of running-difference EUVI/B images in the area of the oscillating loop during the time period of 2007 Jun 26, 17:56:00 UT and 18:26:00 UT. The amplitude measurement of the oscillating loop is carried out along the cross-sectional slit marked with a diagonal bar (from Aschwanden et al., 2009bJump To The Next Citation Point).
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Figure 33: Time-slice plots (color) and amplitude of kink mode as a function of time with fitted damped sine function (graphs) for both STEREO/A and B spacecraft, for the loop oscillation event of 2007 Jun 27, 17:30 UT (from Verwichte et al., 2009Jump To The Next Citation Point).

In a subsequent study (Aschwanden, 2009cJump To The Next Citation Point), stereoscopic triangulation has been applied to the oscillating loop and the reconstructed 3D geometry is shown in Figure 34Watch/download Movie: The XY-projection in the image plane of STEREO/A and B is rendered in the left-hand panels, and the YZ-projections as viewed from the east direction at the east limb are shown in the right-hand panels. The spacecraft separation angle at that time was αsep = 8.26°. The triangulation has been repeated three times with independent manual tracings and the three solutions are shown in Figure 35View Image, projected parallel and perpendicular to the average loop plane. Surprisingly, the triangulated loop reveals an S-shaped geometry in the vertical projection top-down (Figure 35View Image, right panels), and a heavily deformed asymmetry in the side-on projection (Figure 35View Image, middle panels). Thus, the oscillating loop seems to be neither circular nor coplanar. This is an intriguing new result that could not have been obtained without STEREO observations.

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Figure 34: mov-Movie (1131 KB) 3D reconstruction of the same oscillating loop as Figure 32View Image, observed on 2007 Jun 27, 18:19 UT, superimposed on a highpass-filtered image of STEREO/A (bottom) and STEREO/B image (top), using the stereoscopic triangulation method. The loop shape is traced with 9 points (red crosses), interpolated with a 2D spline (red curve), and fitted with an elliptical geometry (yellow curve). The circular model (yellow curve) and the solution of the 3D reconstruction (red curve) is also projected into the (z,y) plane (right panels), with z being the LOS (from Aschwanden, 2009c).
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Figure 35: Projections of the reconstructed 3D loop geometry (see Figure 34Watch/download Movie) into the 3 orthogonal planes of the loop coordinate system: edge-on (left), side-on (middle), and top-down (right). The three rows present 3 independent trials of manual loop tracings.

If we fit a circular geometry to the same 2D loop tracings (dashed yellow curves in Figure 34Watch/download Movie), we obtain from the average and standard deviations of 3 independent tracings the following parameters: a baseline azimuth angle of ∘ ∘ α = 27.0 ± 4.5, a loop plane inclination angle of ∘ ∘ 𝜃 = 24.1 ± 1.0, a loop radius of r = 83 ± 10 Mm, and a full loop length of L = 319 ± 19 Mm. The full 3D loop length of the triangulated loop is slightly shorter than the circular model, i.e., L = 311 ± 22 Mm. This compares favorably with the semi-circular forward-fitting of Verwichte et al. (2009), who obtained values of α = 19∘, 𝜃 = 27∘, r = 110 Mm, and L = 346 Mm. Thus, we conclude that the circular forward-fitting model yields values that are very close to those of the stereoscopic method, differing by ∘ <∼ 10 for the loop plane inclination angle and for the loop length, which is the most important parameter for coronal seismology. However, the forward-fitting model makes the assumption of circularity and coplanarity and, thus, cannot reveal asymmetric and non-planar shapes of the oscillating loop.

In Figure 35View Image, we transformed the loop coordinates from the image coordinate system (xA, xB, xZ) into the loop coordinate system (X, Y, Z), which is defined in the following way: the horizontal X-axis is aligned with the loop baseline between the two loop footpoints, with the midpoint as center X = 0. The vertical Z-axis intersects the baseline midpoint and the apex midpoint of the loop, while the Y-axis is perpendicular to the X − Z plane. The coordinate transformation is accomplished by 3 subsequent rotations (i.e., by the azimuth angle α, the loop plane inclination angle 𝜃, and the longitude angle l1, see Figure 23View Image). The oscillating loop shown in Figure 35View Image is clearly heavily deformed and deviates significantly from a coplanar and circular model. The maximum deviations Δr from the average loop radius r are 0.54 < (Δr ∕r) < 1.45, and the non-planarity reaches up to <∼ 0.21 of the loop radius, which is much larger than for typical active region loops, which have radial variations of <∼ 1.2 and non-planarity deviations of <∼ 0.1. Obviously, this loop is highly non-circular and non-planar, possibly affected by the dynamic magnetic forces associated with the launch of a CME that usually is responsible for triggering loop oscillations.

Using the method of stereoscopic 3D triangulation, as shown for a single time frame at 2007 Jun 27, 18:18:30 UT in Figure 34Watch/download Movie, we repeat now the same procedure for 12 (epipolar-coaligned) stereoscopic image pairs in the time interval of 17:58 – 18:26 UT. The same time sequence is shown in form of running-time difference images in Figure 32View Image, yielding an oscillation period of P = 565 s and an exponential damping time of τdamp ≈ 1600 s. The triangulated 3D geometry of the oscillating loop is shown in Figure 36Watch/download Movie, which has a consistent asymmetric S-shape throughout the entire observed time interval (≈ 35 min) as shown in Figure 34Watch/download Movie, while the oscillation amplitude is much smaller than the deviations of the loop geometry from a circular model. We display the average amplitude of loop motions in x-direction (east-west amplitude dx(t) in Figure 36Watch/download Movie top right panel), and in z-direction (line-of-sight amplitude dz (t) in Figure 36Watch/download Movie middle right panel). Both amplitudes show a correlated oscillation with similar periods and amplitudes, i.e., P = 681 s for dx (t) and P = 685 s for dz(t), and A0 = 1.4 Mm for dx (t) and A0 = 1.5 Mm for dz (t), respectively. The amplitude dx (t) corresponds to an oscillation in the vertical plane (with respect to the solar surface), and dz (t) to an oscillation in the horizontal plane, but the two oscillations are almost in phase (t ∕P = 0.19 0 vs. t∕P = 0.26 0). The fact that the two oscillation directions have a similar amplitude implies that the polarization is a combination of horizontal and vertical planes with a polarization angle near 45°. However, since the average 3D loop geometry corresponds to a non-planar, helically-twisted shape, the 3D motion can also be described in terms of circular polarization or a torsional mode. It is instructive to inspect a movie visualization of Figure 36Watch/download Movie.

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Figure 36: mov-Movie (856 KB) 3D reconstruction of loop oscillations for a sequence of 16 EUVI/A+B 171 Å images in the time interval of 2007 Jun 27, 17:58 – 18:26 UT, using the stereoscopic triangulation method. The loop tracings in EUVI/A are rendered in the x-y plane (bottom left panel), while the orthogonal reconstruction are shown in the x-z plane (top left panel) and in the z-y plane (bottom right panel). The loop tracings are rendered with grey curves, the semi-circular fit with a dashed curve, and the curvature radius maximization method with a thin black curve. The oscillation amplitudes averaged in the loop segments 0.3 < s ∕L < 0.6 (marked with thick black curves) are shown in x-direction (east-west amplitude dx(t) in top right panel) and in the z direction (line-of-sight amplitude dz(t) in middle right panel).

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Figure 37: mov-Movie (831 KB) Orthogonal projections of a triangulated oscillating loop (frames in top half) during the same time interval as shown in Figure 13View Image (with the time marked by colors, progressing in order of brown-red-orange-yellow). The bottom panels visualize the same projections of a helically twisted model loop (see model parameters in text).

In Figure 37Watch/download Movie we show the triangulated loop motion projected into the 3 orthogonal directions of the loop coordinate system. In order to illustrate the plausibility of circular polarized kink motion we simulate a helically twisted loop (with a twist of 0.75 turns and geometric tapering of the toroidal radius towards the footpoints) that undergoes a periodical change in the twist angle, where the displacement is highest at the loop apex and falls off towards the footpoints. This simple simulation shown in Figure 37Watch/download Movie (bottom half) demonstrates qualitatively how the observed loop shapes and motions (Figure 37Watch/download Movie top half) can be explained by a circular polarization. Some further modeling studies are required to decide whether we deal with a circularly polarized (helical) kink mode or with a torsional wave mode. Torsional wave modes and helical loop geometries are of particular interest for triggering mechanisms of flares and CMEs by the kink instability. A large twist angle of Φ = 12 π was recently observed to trigger a flare (Srivastava et al., 2010). In addition, numerical MHD simulations with realistic initial conditions for the excitation mechanisms may shed some light on the coupling of various known MHD wave modes, i.e., kink-modes, sausage-modes, and torsional modes (Ofman, 2009).

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