Multi-instrument observations of a failed flare eruption associated with MHD waves in a loop bundle
Giuseppe Nisticò, Vanessa Polito, Valery M. Nakariakov, Giulio del Zanna
aa r X i v : . [ a s t r o - ph . S R ] D ec Astronomy & Astrophysicsmanuscript no. paper_arXiv c (cid:13)
ESO 2018July 18, 2018
Multi-instrument observations of a failed flareeruption associated with MHD waves in a loopbundle
G. Nisticò ⋆ , V. Polito , V. M. Nakariakov , and G. Del Zanna Centre for Fusion, Space and Astrophysics, Department of Physics, University of Warwick,Coventry CV4 7AL, United Kingdom Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cam-bridge CB3 0WA, United KingdomJuly 18, 2018
ABSTRACT
Context.
We present observations of a B7.9-class flare that occurred on the 24th January, 2015,using the Atmopsheric Imaging Assembly (AIA) of the Solar Dynamics Observatory (SDO),the EUV Imaging Spectrometer (EIS) and the X-Ray Telescope of Hinode. The flare triggersthe eruption of a dense cool plasma blob as seen in AIA 171Å, which is unable to completelybreak out and remains confined within a local bundle of active region loops. During this process,transverse oscillations of the threads are observed. The cool plasma is then observed to descendback to the chromosphere along each loop strand. At the same time, a larger di ff use co-spatialloop observed in the hot wavebands of SDO / AIA and Hinode / XRT is formed, exhibiting periodicintensity variations along its length.
Aims.
The formation and evolution of magnetohydrodynamic (MHD) waves depend upon thevalues of the local plasma parameters (e.g. density, temperature and magnetic field), which canhence be inferred by coronal seismology. In this study we aim to assess how the observed MHDmodes are a ff ected by the variation of density and temperature. Methods.
We combined analysis of EUV / X-ray imaging and spectroscopy using SDO / AIA, Hin-ode / EIS and XRT.
Results.
The transverse oscillations of the cool loop threads are interpreted in terms of verticallypolarised kink oscillations. The fitting procedure provides estimates for a period of ∼ ∼ ⋆ Now at George-August-Universität Göttingen, Germany (e-mail: [email protected] ). Article number, page 1 of 23isticò et al.: A failed flare eruption associated with MHD wavesin agreement with the density variations due to the presence of the plasma blob inferred fromthe intensity light curve at 171Å. The coexisting intensity oscillations along the hot loop areinterpreted as a slow MHD wave with a period of 10 min and phase speed of approximately436 km s − . Comparison between the fast and slow modes allows for the determination of theAlfvén speed, and consequently magnetic field values. The plasma- β inferred from the analysisis estimated to be approximately 0.1–0.3. Conclusions.
We show that the evolution of the detected waves is determined by the temporalvariations of the local plasma parameters, caused by the flare heating and the consequent cooling.We apply coronal seismology to both waves obtaining estimations of the background plasmaparameters.
Key words.
Sun: corona – Sun: MHD waves – Techniques: spectroscopic
1. Introduction
Observations of magnetohydrodynamic (MHD) waves in the solar corona provide us with an im-portant tool for the determination of the local plasma parameters using seismology (e.g. Robertset al. 1984). The advent of space observatories during the last two decades has increased the rich-ness of MHD wave phenomena found in the structured medium of the solar corona at extreme ultraviolet (EUV) and X-rays wavelengths, encompassing di ff erent spatial and temporal scales. Morerecently, observations at even higher resolution (1 ′′ , 12s cadence) with the Atmospheric ImagingAssembly (AIA) on board the Solar Dynamics Observatory (SDO) (Lemen et al. 2012) have sub-stantially contributed to improving our view and knowledge of MHD modes in coronal structures.MHD waves have been observed as large-scale disturbances propagating through the solar disk,also known as EUV global waves (e.g. Patsourakos & Vourlidas 2012); transverse oscillations(kink waves) of coronal loops (Aschwanden et al. 1999; Nakariakov et al. 1999), prominences(Hershaw et al. 2011); propagating and standing longitudinal oscillations (slow waves) in loopsand polar plumes (DeForest & Gurman 1998; Wang et al. 2007; Kiddie et al. 2012; Krishna Prasadet al. 2012) and quasi periodic fast wave trains in coronal funnels (Liu et al. 2012; Nisticò et al.2014). These observations have confirmed the theory of MHD modes in a cylindrical magneticflux tube (e.g. Edwin & Roberts 1983; Roberts et al. 1984), which is taken as the basic model todescribe dynamics of a field-aligned non-uniformity of the plasma density, typical for the corona.Kink oscillations of coronal loops have received particular interest due to their abundance in thesolar corona, which allows us a systematic application of coronal seismology, and to quantify theirpossible contribution to coronal heating (e.g. Goddard et al. 2016). Indeed, after being triggered bylocal coronal eruptions (Zimovets & Nakariakov 2015), the wave amplitude is observed to decayexponentially in a few cycles, which is believed to be caused by resonant absorption.In addition to this scenario, recent studies show the existence of a further class of kink oscilla-tions characterised by the absence of damping (Wang et al. 2012; Nisticò et al. 2013a; Anfinogentov Article number, page 2 of 23isticò et al.: A failed flare eruption associated with MHD waves et al. 2013, 2015). The application of coronal seismology to kink oscillations of coronal loops hasvery recently received a further incentive with the discovery of a Gaussian profile at the early stageof the decay trend (Hood et al. 2013), providing a new method to uniquely determine the densitycontrast and the inhomogeneous layer width in coronal loops (Pascoe et al. 2016).A combination of EUV imaging with additional observational techniques, for example in theradio band or EUV spectroscopy, provides us with further constraints on the determination of thecoronal plasma quantities (e.g. Kim et al. 2012; Kupriyanova et al. 2013; Verwichte et al. 2013).Indeed, for example, parameters such as the adiabatic index γ and the molecular weight µ , whichare present in the definitions of the sound and Alfvén speeds, are usually assumed to standard valuestypical for the corona ( γ = / µ = . ff ective adiabatic index is not equal to the assumed value of 5 /
3. Furthermore, observations ofcoexisting di ff erent MHD modes in the same coronal structure can provide compelling constraintson the determination of these plasma quantities (Van Doorsselaere et al. 2011a; Zhang et al. 2015).In this paper, we present a multi-instrument analysis of a flare which has been observed onthe 24th January, 2015, using SDO / AIA, the EUV Imaging Spectrometer (EIS; Culhane et al.2007) and the X-Ray Telescope (XRT; Golub et al. 2007) onboard the Hinode satellite, launchedin 2006. The flare triggers an eruption of a dense and cool plasma blob, driving kink oscillationsin nearby cool loop threads, and forms a di ff use hot loop, which in addition exhibits longitudinaloscillations of the EUV intensity interpreted as a slow magnetoacoustic wave. The aim of thisstudy is to assess the evolution of the two MHD modes observed in association with the observedvariation of density and temperature. The paper is structured as follows. In Section 2 we describethe observation and the instruments; in Section 3 we discuss the investigation of the 3D structureof the loop; in Section 4 and 5 we present the analysis of the kink and slow waves, respectively.Discussion and a conclusion are given in Section 6.
2. Observations and data analysis
The eruption that we have analysed occurred in the active region NOAA 12268 and is associatedwith a B7.9-class flare as recorded by the GOES satellite, which measured a peak in the X-rayflux at 12:00 UT on the 24th January, 2015. To study the event, we used data from SDO / AIA,Hinode / EIS and XRT.
SDO / AIA produces full-disk images of the Sun in seven EUV wavelength bands (as well as in theUV) with a cadence of 12 s, a nominal pixel size of 0.6 ′′ and a spatial resolution of approximately1 ′′ . We downloaded one hour of observations in all the EUV channels of AIA between 11:50 and12:50 UT and processed them using the standard SolarSoft program aia_prep.pro . Figure 1 showsthe active region observed in di ff erent AIA EUV wavebands and an X-ray image with the Be_thinfilter of the XRT telescope. The AIA EUV images are excellent for their resolution, but, with the Article number, page 3 of 23isticò et al.: A failed flare eruption associated with MHD waves
Fig. 1.
SDO / AIA and Hinode / XRT images of the AR NOAA 12268 on the 24th January, 2015, at the time ofthe flare event. The box in the 304 Å image shows the Hinode / EIS FOV. The labels indicate: a) a small post-flare loop at the site of the initial eruption; b) an expanding blob driving kink oscillations in the neighbouringloops; c) a bright loop observed distinctly at 131 and 94 Å; and d) overlaying unperturbed loops. The timeevolution in the 171 and 94 Å as well as a composite of di ff erent channels is shown in a movie availableonline. The composite contains the bands 304 (red), 171 (green) and 94 Å (blue). The field-of-view of themovie is shown in the figure as a dotted square. exception of the 171 Å band, are strongly multi-thermal, as described in O’Dwyer et al. (2010);Del Zanna et al. (2011); Petkaki et al. (2012); Del Zanna (2013b), for example. Assessing theplasma temperature from the AIA images is therefore prone to some uncertainties, and it is onlyby combining the AIA information with that from the EIS and XRT that we are able to understandthe temperature evolution of this complex event. At the time of the observations, Hinode / EIS was pointing at this active region, even though therewas a limited field of view (FoV), as shown in Fig. 1. The EIS observations consisted of a sequenceof ‘sparse’ raster studies
HH_Flare_raster_v6 from approximately 06:00 UT to 15:30 UT. Eachraster contained 20 × ′′ slit steps covering an area of 60 ′′ × ′′ in ≈
212 s. The exposure timewas ≈ eis _ prep . pro , including standard options as described in the EIS software notes . In orderto convert the data number (DN) to physical units, we applied the radiometric calibration methoddescribed in Del Zanna (2013a), which accounts for the degradation of the detectors’ e ffi ciencyover time. EIS data were also corrected for the o ff set (approximately 18 pixels) between the longwavelength (LW) and short wavelength (SW) CCD channels, and the wavelength tilt along the slit . http: // solarb.mssl.ucl.ac.uk:8080 / eiswiki / Article number, page 4 of 23isticò et al.: A failed flare eruption associated with MHD waves
Fig. 2.
Images of SDO / AIA 304 (a) and 131 Å (b), Hinode / EIS He II (c, T ∼ ∼ EIS observes emission lines formed over a broad range of temperatures, from the cool He II linewith a formation temperature of ≈ T ≈ ff erences. The green squares mark the locations where the plasma densityhas been estimated (values are in cm − ) from the intensity ratio of the EIS Fe XIV 264.79 and274.20 Å lines (see Sect. 2.4). The Fe XIV line ratio provides useful electron density diagnostics inthe 2 MK plasma. We used atomic calculations from the CHIANTI v.8 database (Del Zanna et al.2015). The Fe XIV line at 264.79 Å is free of significant blends (Del Zanna et al. 2006), whilstthe Fe XIV 274.20 Å is blended with a Si VII line at 274.17 Å. However, during flares, the SiVII line usually contributes only approximately 4 % of the intensity of the Fe XIV 274.203 Å line(Del Zanna et al. 2006; Brosius 2013). The Fe XIV ratio therefore provides a lower estimate of theelectron density of the plasma. Article number, page 5 of 23isticò et al.: A failed flare eruption associated with MHD waves
Table 1.
Timeline of the event referring to the SDO / AIA 171 images.
Time (UT) SDO / AIA11:51:48 First brightenings.11:57:36 Increase in the brightness in the active region.12:00:00 Uplift of a plasma blob in 171 Å and formation ofa hot loop in 131 and 94 Å.12:02:00 The plasma blob hits some loops and remains con-fined, driving kink oscillations.12:11:36 The blob di ff uses along the loop threads, kink os-cillations continue. Longitudinal intensity oscilla-tions in 94 Å.12:20:00 The cold plasma is observed to descend along theloop threads. XRT acquires images of the full-Sun in several X-ray broad-band filters. In this work, we useimages formed in the Be_thin and Al_poly filters, which are available for the whole duration of theevent under study, and which have similar cadences ( ≈ xrt_prep.pro routine, which removes the telescope vignetting(Kobelski et al. 2014) and subtracts the dark current from the detector’s signal. The XRT filters arebroad-band and highly multi-thermal, as described in O’Dwyer et al. (2014a).For an isothermal plasma, the ratio of di ff erent XRT bands can provide reliable temperaturemeasurements (Narukage et al. 2014; O’Dwyer et al. 2014b). The top-right panel of Fig. 3 showsthe temperature map obtained by using the ratio of the XRT Be_thin and Al_poly images at 12:04UT, close to the peak of the flare. The temporal evolution of the temperature averaged over theboxcars 1 (green) and 2 (light blue) is given in the bottom-left panel of Fig. 3. See also the movieattached to Fig. 3 to follow the evolution of the temperature map over time. The theoretical re-sponse of each filter was obtained by convolving the filter e ff ective area with an isothermal spectraproduced by CHIANTI version 8 using chemical abundances from Asplund et al. (2009), ionisationequilibrium calculations from Dere et al. (2009) and a plasma density of 10 cm − . We have veri-fied that using di ff erent values for the plasma density does not a ff ect the theoretical XRT responsefunction (for densities of 10 and 10 cm − , the functions almost coincide), and only a muchhigher density can modify the observed spectrum by each XRT filter significantly. An estimate ofthe plasma density for the hot loop is also obtained from the emission measure, after having de-fined the average temperature and a typical column depth of ∼ ′′ . Figure 3-bottom right shows theevolution of the density averaged over the boxcars 1 and 2, respectively. The density values rangebetween 3 × cm − and 6 × cm − . The movie attached to Fig. 1 shows the evolution of plasma structures in the event, while Table 1summarises a timeline of the event as observed in the AIA 171 Å channel. The event is very rich infeatures. Indeed, a very bright small post-flare loop appears in di ff erent AIA wavebands (131, 211, Article number, page 6 of 23isticò et al.: A failed flare eruption associated with MHD waves
Fig. 3.
Top: images (reversed colours) of Hinode / XRT from the Be-thin (left) and the Al-poly (middle) filters,and temperature maps obtained using the ratio of the two filters (right). The temporal evolution of the temper-ature map is shown in a movie available online. The entire hot loop bundle has a temperature of 6–10 MK.Bottom-left: temporal evolution of the plasma temperature at the position indicated by the boxcars 1 (green)and 2 (light blue) in the top-right panel. Bottom-right: temporal evolution of the density in the hot loop fromthe boxcars 1 and 2.
335 and 94 Å, see feature a) in Fig. 1). This structure is also clearly visible in the EIS Fe XIV–FeXV and Fe XVI lines, indicating that the plasma is mostly emitting at approximately 2–3 MK.At the footpoint of this small flare loop, we measure densities of approximately 1.7 × cm − using the EIS Fe XIV line ratio (as indicated in the top panel of Fig. 2), in contrast to a backgrounddensity of ∼ × cm − .From the region where the small post-flare loop forms, we observe an eruption of a brightplasma blob, which is clearly visible in the 304 and 171 channels and may be assimilated to asmall flux rope. The bulk of this plasma has chromospheric / transition region (TR) temperatures,because it shows strong emission in the EIS He II and other TR lines. We also note that this eventwas partially observed by IRIS (De Pontieu et al. 2014). The IRIS slit-jaw images in the Si IVfilter clearly show the eruption of this filamentary cool (T ≈ − .A few minutes later (after the flare peak), at 12:04 (bottom panels of Fig. 2), this blob becomesvisible only in He II. Indeed, Fig. 2 shows that there is no Fe XIV (2 MK) emission. On the otherhand, the small loop (feature a) in Fig. 1) is strongly emitting in Fe XIV and the plasma density atthe top of the loop is now 1 . × cm − (as indicated in the bottom panel of Fig. 2).The cool plasma expands until it hits some overlaying loops, triggering kink oscillations, andremaining confined within them (see b) in Fig. 1). These kink oscillations are only visible with Article number, page 7 of 23isticò et al.: A failed flare eruption associated with MHD waves
AIA (they were mostly outside the IRIS and EIS FoV, see Fig. 1). Finally, this cool plasma isobserved to descend back to the chromosphere flowing along several loop threads in the form ofcoronal rain. This process lasts for approximately 20 min. The downflow of this cool plasma onthe western side is observed in the IRIS data and in the EIS He II and TR lines (O V, Fe VIII).The apparent downflow speed along the loop threads is of the order of 10 km s − . The observedphenomenology would be of interest for comparison with modelling by Oliver et al. (2014, 2016),which is out of the scope of this work. A bright and di ff use loop in feature c) is seen in the 131and 94 Å wavebands, while the larger loops observed in 171, 193 and 211 Å and overlaying theactive region (marked by feature d) in Fig. 1) do not show significant oscillations and are almostunperturbed.In contrast to the fine loop threads observed with the AIA 171 Å, the sequence of the AIA94 Å images shows the formation and evolution of a well-defined di ff use hot loop, adjacent to thecool filamentary material and exhibiting longitudinal intensity oscillations (see the movie attachedto Fig. 1). This di ff use loop appears physically distinct from the nearby loop threads observed inthe ‘cooler’ AIA channels. While the loop threads in 171 Å are very narrow with approximatelyconstant width ( ∼ ff ect the dynamics of the transverse and longitudinal waves.
3. 3D geometry
In the context of MHD and coronal seismology (Nakariakov et al. 2016), it is important to deter-mine the 3D structure of coronal loops, that is, their full (and not projected) length, the inclinationand azimuthal angles (Nisticò et al. 2013b) in order to correctly interpret the periodic intensityvariations, which are modulated by the periodic changes of the column depth (Cooper et al. 2003),and to unambiguously identify the polarisation of kink oscillations. Inference of the 3D geometryby stereoscopy in this case is impossible due to the lack of data from the Solar Terrestrial RelationsObservatory (STEREO), since the two spacecraft were in the back hemisphere of the Sun at the
Article number, page 8 of 23isticò et al.: A failed flare eruption associated with MHD waves time of this observation. However, under some assumptions, it is possible to obtain a reliable curvethat fits the series of points, which are manually determined and sample the bundle of threads ob-served in AIA 171 (red dots) and the di ff use loop in AIA 94 (yellow dots) (see Fig. 4). We adoptthe technique described in Verwichte et al. (2010). Fig. 4.
Three-dimensional reconstruction of the bundle of loop threads (red dots) observed with the AIA171 Å (top panels), and the hot loop (yellow dots) observed with the AIA 94 Å (middle panels). The loops arebest-fitted by semi-ellipses (solid green line) with an inclination angle of -60 deg with respect to the normalsurface, and heights of 0.85 r L in the 171 Å, and 0.70 r L in the 94 Å channels. The projections of the loopsystem for di ff erent orientations of the HEEQ coordinate system are shown in the bottom panels. The 3D structure of the loop bundle was determined by initially considering a semi-circle in areference frame defined by three orthogonal axes: the loop baseline e b (red arrow), the normal tothe solar surface e n (green arrow) and the vector e t = e b × e n (blue arrow). The footpoints have Article number, page 9 of 23isticò et al.: A failed flare eruption associated with MHD waves
Stonyhurst longitude and latitude equal to (-61.6,-8.6) and (-53.4,-8.7) deg, respectively. The loopcentre is consequently found as the mean between the footpoints, and has coordinates (-57.5,-8.7)deg. The footpoint half-distance represents the radius of the semi-circle, which is approximately 50Mm. We try to match the model with the observations by varying the inclination angle θ measuredwith respect to the solar surface between -90 and +
90 deg. The left-top panel of Fig. 4 shows theloop as a semi-circle ( h L = r L ). For simplicity we show only three cases for θ = -90 (dashed), -60(continuous), and -30 deg (dashed green line). The white arrows represent the loop height for thedi ff erent inclinations. A good approximation is obtained for θ = −
60 deg. This estimate is furtherjustified by the measurements of the speed for the plasma blob, which is assumed to move alongthe loop plane. Indeed, as shown in the previous section, the Doppler shift, which is assumed tobe related to the plasma blob expansion, defines the speed along the line-of-sight v ⊥ ≈
100 kms − , while the projected speed on the plane of sky is estimated as v k ≈ −
178 km s − (see nextsection). The loop inclination angle can then be found as θ ≈ tan − ( v k /v ⊥ ) = −
60 deg. We varythe loop height to improve the fit ( h L = (1 . , . , . , . , ... ) r L , where r L is the major radius).The best fit is visually found for h L = . r L , therefore the loop seems to be slightly elliptical.However, the right loop leg is not adequately reproduced, maybe because of the presence of somefurther tilt in the loop plane orientation, or the departure from the plane shape, that is, a sigmoidshape. The bright and di ff use emission observed in the hot channels appears to be located slightlylower than the cooler threads, even if this statement maybe rather subjective. The loop shape inthis case is sampled by the yellow dots in the central panels of Fig. 4. The points are fitted with acurve with a height of 0 . r L (green line) . Given the 3D orientation, we can project the series ofpoints onto the loop plane as clearly described in the appendix of Verwichte et al. (2010). The looplength is found by summing the distances along these points. Therefore, assuming an uncertaintyof the measure of 10%, for the cool threads observed in the 171 channel, we obtain the length L = ±
14 Mm, (which is comparable with that of the best-fitting ellipse, ∼
144 Mm), whilefor the hot loop, the length is L = ±
13 Mm (the best-fitting ellipse with an height of 0.70Mm has a length of 133 Mm). In addition, we have qualitatively investigated the polarisation ofthe transverse oscillations of the cool loop threads. Having defined the loop geometry (radius andheight) and its oscillatory dynamics (amplitude, period and damping), the right panels of Fig. 5show the model for a loop in a 3D Cartesian coordinate system at the equilibrium (green line)and at the extrema of the oscillations (blue dashed lines) for the vertical polarisation (top panel)with the motion strictly on the xz plane, and the horizontal polarisation (bottom panel) where themotion takes place on the x y plane. A comparison of the oscillation modes with the observationsat 171 Å (left panels) reveals that vertically polarised kink oscillations may match the transversedisplacements of the observed loop threads (see the movies attached to Fig. 5). Article number, page 10 of 23isticò et al.: A failed flare eruption associated with MHD waves
Fig. 5.
Top: vertical polarisation for the transverse displacements of the coronal loop threads. The right panelshows the model for a loop at the equilibrium (green line) in a 3D Cartesian coordinate system. The distanceis measured in loop radius r L units. The shape of the loop at the oscillation extrema is represented by thedashed blue lines. The motion is on the xz plane. The left panel shows the projection of the loop model andits configuration at the oscillation extrema projected in the AIA 171 FoV at the time of the start of the kinkoscillations. The red dots sample a single loop thread at the equilibrium, which is fitted by the loop model ingreen, and the red diamond is the loop centre on the solar surface. The movement direction of the loop modelat the oscillation extrema matches that of the blob and the overall loop threads. Bottom panels: the same asdescribed above but for the case of the horizontal polarisation mode. In this case, the motion is strictly on the x y plane (right panel), and the loop configuration at the oscillation extrema does not fit the observations well(left panel). Animations of the top and bottom panels are available in the two online movies.
4. Kink oscillations of the cool threads
To analyse the transverse oscillations of the loop threads, we have selected two slits, S1 and S2,directed as in the left panel of Fig. 6. From these slits we have extracted the intensity for each frameof our dataset and constructed time-distance (TD) maps. The loop strands are clearly visible in the171 Å channel of AIA. The TD maps at this wavelength are given in the right panels of Fig.6.In the TD maps, the signature of the expanding blob appears in the form of a very bright andinclined feature or peak. Its slope provides us with an estimate of the projected speed (green pointsin Fig. 6 - right panels). Before the eruption, a much slower expansion with a linear speed of15–17 km s − is measured (blue points). After this phase, the blob is expanding with a projectedvelocity of approximately 178 km s − along slit S1, while in S2 the speed is lower, 134 km s − ,since this direction does not exactly match the one of the expanding blob. The expansion of theblob displaces the loop threads from their equilibrium, which undergo transverse oscillations (redpoints in the TD maps). The patterns in the TD maps are composed of several strands, which arenot very easy to track, and oscillate collectively. The oscillations in the right panels of Fig. 6 are Article number, page 11 of 23isticò et al.: A failed flare eruption associated with MHD waves
Fig. 6.
Left: Image from SDO / AIA 171 Å with the slits S1 and S2 used to make TD maps and trace thetransverse oscillations. Right: TD maps from the S1 (top) and S2 (bottom) slits. tracked by following the upper rim of the oscillating bundle by eye. Therefore, we characterise theoscillatory dynamics of the overall bundle rather than that of each single loop thread.
Fig. 7.
Time series of the oscillations S1 and S2 (left), wavelet power spectra of the time series (centre), andplots of the period vs time (right). The period is between 3.7 and 4 min. There is a very small variation of theperiod from the fitting analysis, however the green lines in the wavelet power spectra have a null slope.
The oscillatory patterns have been fitted using the MPFIT routines (Markwardt 2009) with thefollowing function y ( t ) = y + A cos π ( t − t ) P + P ′ ( t − t ) + φ ! exp (cid:18) − t − t τ (cid:19) , (1) Article number, page 12 of 23isticò et al.: A failed flare eruption associated with MHD waves
Table 2.
List of the fitting parameters for the observed oscillations.
Oscillation t y A P P ′ φ τ [min] [Mm] [Mm] [min] [min min − ] [deg] [min]S1 11.6 0.3 ± ± ± ± ± ± ± ± ± ± ±
11 7.9 ± ± ± ∗ ± ± ± ± Notes.
Kink oscillations S1 and S2 are shown in Fig. 7, while the longitudinal oscillation S3 refers to Fig. 9. ∗ The amplitude of the slow MHD wave S3 is not in Mm units but normalised to the loop length. which assumes a priori a linear dependence of the period on the time, with P ′ being a variationrate of the period. Before applying the fitting routine, the time series was detrended with a back-ground linear fit. The fittings were weighted by the errors of each data point, which was taken tobe approximately 2 pixels ( ∼ P of the oscillations is between 3.7 and 4.0 min. Theperiod rate change P ′ is negative in both cases (even if the standard deviations associated to theseestimates make them relatively insignificant), and is consistent with a decrease of the density, aswe will see in more detail in the next subsection. The wavelet power spectra of the time series inthe central panels of Fig. 7 shows that the period of the kink oscillations is approximately 3.5 min(green line). The transverse displacements observed in the loop are interpreted in terms of the fundamentalstanding fast magnetoacoustic kink wave. The phase speed V (K)ph is determined by the loop lengthand the period of the oscillation, that is, V (K)ph = LP . (2)In this case, given the length L = L = ±
14 Mm and the period of P = . ± . V (K)ph = ±
152 km s − (see also Eqs. A.1-A.2 in Appendix A). From theoreticalmodelling of MHD modes in a plasma cylinder (e.g. Edwin & Roberts 1983), the phase speed V (K)ph for long-wavelength kink oscillations (in comparison with the minor radius of the loop) is thekink speed C K , which is the density-weighted average of the Alfvén speeds inside and outside theoscillating plasma cylinder: C K = ρ C + ρ e C ρ + ρ e / . (3)In the low- β plasma regime, typical for coronal active regions, the expression above can be approx-imated as C K ≈ C A + ρ e /ρ ! / , (4) Article number, page 13 of 23isticò et al.: A failed flare eruption associated with MHD waves where C A = B / p πρ is the Alfvén speed, and ρ is related to the number density n with therelation ρ = µ m p n . Therefore, changes in density can a ff ect the values of C K , and consequentlythe period P , as also found in the decayless oscillation event described by Nisticò et al. (2013a).Indeed, using the expressions above it is relatively straightforward to show that P = LB p πρ (1 + ρ e /ρ ) . (5) Fig. 8.
Intensity time series in the AIA 171 Å band (bottom) starting at 11:50 UT, obtained from a boxcarcentred on the cool plasma blob (top). The intensity shows a strong peak at the flare peak time, and then itdrops very quickly in approximately 5 min. In the inset plot we show that the oscillations in the intensityprofile (black) are in antiphase with the loop thread displacement (dashed red line).
After its expansion, the blob is observed to slowly di ff use and descend along the loop threads.The observed longitudinal flow could also be induced by the ponderomotive force associated withthe nonlinear kink oscillation (e.g. Terradas & Ofman 2004), but the theory of this e ff ect is notelaborated enough to make any quantitative comparison. Perhaps the weakly compressive natureof long-wavelength kink oscillations would allow one to adopt the results obtained for the pon-deromotive force in Alfvén waves (e.g. Tikhonchuk et al. 1995; Verwichte et al. 1999; Thurgood &McLaughlin 2013). To investigate the influence of the downflowing plasma blob on the period ofthe kink oscillations, we analyse the intensity time series from the SDO / AIA 171 Å band. Figure 8shows the intensity time series I averaged over a boxcar centred on the cool blob (bottom panel). Article number, page 14 of 23isticò et al.: A failed flare eruption associated with MHD waves
The profile resembles that of a shock with a very sharp ramp (at ∼ ∼
15 min). Since I ∝ n , then variations of the intensityon time-scales larger than the kink period can be expressed as δ I / I = δ n / n , assuming that theplasma does not dramatically change its temperature, and the angle between the local loop segmentand line-of-sight remains constant, as the observations suggest.On the other hand, from Eq. 5, the variations of the period with respect to the density (con-sidering the inner density ρ or equivalently n ) are δ P / P = δ n / n . Therefore, P changes withrespect to I as δ P / P = δ I / I . For δ I = −
50 DN in a time interval, ∆ t =
10 min and I = δ P / P ≈ − .
03. By taking P = . δ P ≈ − . P = − .
11 min, andtherefore P ′ ≈ δ P / ∆ t = − .
01 min min − , which is consistent with the values obtained from thefittings. It is worth mentioning that the oscillations that are seen in the intensity profile (see the insetplot in Fig. 8-bottom panel) are in anti-phase with the displacement of the loop threads. This can beexplained in terms of vertically polarised oscillations of a bundle of loop threads as suggested byAschwanden & Schrijver (2011). In general, such variations are determined by the periodic changein the column depth (Cooper et al. 2003) and the correct polarisation mode can be inferred by ap-propriate forward modeling, as discussed in Verwichte et al. (2009) and recently shown by Yuan& Van Doorsselaere (2016a,b). We can now estimate the Alfvén speed and the magnetic field (seeEqs. A.3–A.6). We considered the following values: L = L = ±
14 Mm, P = . ± . µ ≈ . m p = . × − g, n = (1 . ± . × cm − and n e = (3 . ± . × cm − (wehave assumed uncertainties for the densities of 50%). We note that the density of the cool threadsis very di ffi cult to measure. Because of its lower spatial resolution with respect to SDO / AIA, Hin-ode / EIS observed the loop as a single thread only. The density estimate for n has been obtainedfrom the emission measure, assuming a column depth of 3 ′′ , photospheric abundances and volumefilled. We obtain an Alfvén speed C A = ±
146 km s − , and a magnetic field of B = ±
12 G.A summary of the coronal parameter estimates is given in Table 3.
Table 3.
List of the physical parameters for the observed fast and slow MHD waves.
Wave
L P T n e n V ph C S C A B [Mm] [min] [MK] [10 cm − ] [10 cm − ] [km s − ] [km s − ] [km s − ] [Gauss]Kink 141 ±
14 3.9 ± ±
152 - 972 ±
146 50 ± ±
13 9.7 ± ± ±
47 416 ±
52 - -Slow-tube 10.0 ± ±
58 1255 ±
207 50 ± Notes.
The loop length L , the period P , the temperature T , the densities n e and n are estimated from theobservations and are used as input values for the determination of the sound and Alfvén speeds. The magneticfield is estimated via coronal seismology. In the slow-tube wave approximation, the inner density n of the hotloop is found from the Alfvén speed and the value of magnetic field, which has been previously inferred fromthe kink wave. Article number, page 15 of 23isticò et al.: A failed flare eruption associated with MHD waves
5. Longitudinal oscillations in the hot loop
We recall (see Fig. 3) that the hot loop increases its brightness in the Be_thin and Al_poly filtersand has a temperature of approximately 8 MK. The high-cadence of the AIA instrument allowsus to observe periodic intensity variations along the hot loop in the 94 Å band. Indeed, after theflare, the brightness of the loop seems to vary periodically along the loop axis, bouncing betweenthe footpoints (similar to the observations reported in Kumar et al. 2015). We have considered acurved slit along the loop (red curve in Fig. 9 - top) to extract the intensity along it from eachframe and make a TD map (middle panel in Fig. 9). An oscillatory pattern is clearly visible. Toinvestigate the variation of the period, we have fitted the time series with Eq. 1. The amplitude ofthe oscillation is almost half of the total loop length (0.4 L ) and the initial period P is approximately10 min. The period variation per unit of time is P ′ = .
05 min min − , which is smaller than thatestimated from the wavelet power spectrum (Fig. 9 - bottom). Indeed, the power spectrum exhibitsa clear increase in the period over the time starting from 10 min, with an indicative rate of P ′ = . − (continuous green line in the wavelet power spectrum). The longitudinal oscillation is essentially a slow magnetoacoustic wave, with a phase speed V (S)ph that can be determined from the loop length L = L = ±
13 Mm, and the period P = . ± . V (S)ph = LP = ±
47 km s − . (6)In a low- β plasma, the phase speed can be interpreted as the sound speed of a slow wave, whosemagnitude depends upon the plasma temperature: C S = γ p ρ ! / = γ k B T µ m p ! / ≈ . × − ( γ T /µ ) / km s − . (7)If we consider γ = / µ = .
27 and an average temperature T ∼ ± / XRT, the corresponding sound speed is C S = ±
52 km s − (see also Eqs. A.7-A.8),which is in agreement with the estimated phase speed in Eq. 6. However, the temperature estimatesare a ff ected by strong uncertainties ( ∼ Article number, page 16 of 23isticò et al.: A failed flare eruption associated with MHD waves
Fig. 9.
Top: image of the loop at 94 Å with the curved slit in red. Middle: TD maps from the curved slit. Thered square points track the oscillation and are determined by eye. The oscillation in green is obtained by in-terpolation of the red points. Bottom: wavelet power spectrum of the oscillation profile. The green continuousline shows indicatively the rate at which the period of the slow wave varies (0.1 min min − ), while the greendot marks the value of the period (13 min) after 30 min from its excitation. magnetic flux tubes, and accounting for finite- β e ff ects, the smallest phase speed for a slow MHDmode is given by the tube speed C T = C S C A q C + C . (8) Article number, page 17 of 23isticò et al.: A failed flare eruption associated with MHD waves
For a temperature of T = ± . C S = ±
58 km s − , which is higher thanthe phase speed V (S)ph . Therefore, assuming C T = V (S)ph , we can determine the Alfvén speed for thehot loop (Wang et al. 2007). Indeed, from Eqs. A.9-A.11, and using the estimates for the sound andtube speeds, we find that C A = ±
207 km s − . Assuming a typical magnetic field equal to thatinferred from the kink oscillations, we can obtain an estimate for the density n of the hot loop.Using the relations Eqs. A.12-A.13, we find n = (6 . ± . × cm − , which is in agreementwith the values found with Hinode / XRT (Fig. 3 - bottom right panel). The values of the parametersfrom coronal seismology are summarised in Table 3.It is interesting to note that the variations over time of the period measured from the TD mapand the temperature from Hinode / XRT data are consistent with each other. Indeed, it is easy toshow that δ PP = − δ V (S)ph V (S)ph and δ V (S)ph V (S)ph = δ T T → δ PP = − δ T T , (9)where δ P and δ T are the period and temperature variations in a given time interval, P and T beingthe initial values. We have not considered density variations, which may a ff ect the phase speed viathe dependence of the tube speed on the Alfvén speed. Moreover, the square root dependence ofthe Alfvén speed on the density decreases the e ff ect of the density variation. However, the timeprofile of the density as inferred from Hinode / XRT is almost flat (see Fig. 2). In the range of 30minutes, the period changes from 10 to 13 min (see the wavelet power spectrum in Fig. 9), whilethe temperature drops from ∼ ∼ δ P = δ T = δ P / P = δ T / T = . V (S)ph ≈ C S . For P ∼
10 min and T ∼ L ≈
125 Mm, which is close to our estimatein Section 3.
6. Discussion and conclusion
Observations of fast and slow MHD modes in the same magnetic structure have a crucial rolein coronal seismology. Indeed, in our analysis we show that the values of the magnetic field in-ferred from the observations of the kink oscillations and the coexisting longitudinal slow waveare in agreement. Therefore, it is possible to have a better understanding of the local environmentand justify the robustness of this diagnostic technique (Zhang et al. 2015). In particular, from the
Article number, page 18 of 23isticò et al.: A failed flare eruption associated with MHD waves knowledge of the local sound and Alfvén speeds, the value of the local plasma- β is naturally ensuedas, β = γ C S C A ! . (10)By taking C S = −
465 km s − , C A = − − and γ = /
3, the plasma- β will rangebetween 0.14 and 0.28. On the other hand, the determination of these parameters presumes theknowledge of some other ones, whose values are usually assumed to be known by theory, such asthe adiabatic index γ and the mean molecular weight µ . While the value of µ has less uncertaintiesand is assumed to the standard values of 1.27 from the abundancy of ions in corona (hydrogen andhelium), the local value of γ is subject to discussion. The adiabatic index enters into the definitionof the sound speed and determines the thermodynamics of plasma. Indeed, e ff ective values of γ may di ff er from the theoretical value of 5 / ff ectthe dynamics of the hot loop since the the conduction time is estimated as (p. 321 in Aschwanden2004), τ cond = . × − n T − / L ≈
142 min . (11)In contrast, radiative processes may be important since the temperature of the hot loop decreasesfrom 8 to ∼ ∼
25 min, and the slow wave results to be over-damped (the damping time of theoscillation itself is 40 min as determined by the fitting analysis). Therefore, values of γ are assumedto vary between 1.1 and 5 / ff ective adiabatic index in the solar corona to be γ e ff = . ± . β is very small (see eq. (1) in VanDoorsselaere et al. 2011b) and these perturbations are assumed to propagate like a pure soundwave, which is not the general case in corona. Indeed, if we combine Eqs. 7, 8 and 10, we canexpress the tube speed C T in terms of C S as C T = p + βγ/ C S = γ + βγ/ k B µ m p T ! / . (12)Therefore, interpreting C T as C S can lead to erroneous results for γ if the condition β ≪ γ e ff = γ/ (1 + βγ/ Article number, page 19 of 23isticò et al.: A failed flare eruption associated with MHD waves e ff ective adiabatic index. The relation between β , γ and γ e ff can be written in the following formas, β = γ e ff − γ ! . (13)Assuming a hypothetical value of γ = / ≈ ff erent), the corre-sponding value of γ e ff will decrease from 5 / β , which mayvary through di ff erent coronal regions. It is trivial to show that for β = , γ e ff = γ and, hence, C T = C S . For β = γ e ff = . γ ; this justifies the low- β approximation with the sound speed), but for valuesof β ≥ .
13, the di ff erence becomes remarkable and γ e ff will deviate from the real and unknown γ by more than 10 %. If we use γ e ff = . β ≈ .
6, which is unrealistic for that case: hence, the adiabatic index is surely much lower than 5 / ff erentcurves corresponding to di ff erent values of the adiabatic index γ ranging between the theoreticalvalue of 5 / P =
10 min (the initial periodof our slow wave) with the di ff erent curves identifies the temperature at which the wave exists (for γ = / , . , . , ... , T γ = . , . , . , ... MK). According to the analysis performed with XRT, themaximum temperature reached by the plasma is ∼
10 MK, and consequently the e ff ective adiabaticindex for our observation falls in the range 5 / < γ ≤ .
5. Similarly, we have also considered thetube speed for γ = / γ = . ff erent curves fordi ff erent values of the plasma- β between 0.0 (lower curve in black) and 1.0 (upper curve in purple).For γ = /
3, the possible values of β that fit our observations fall approximately in the range of 0.1to 0.3, while for γ = . , the intersection points are moved towards higher temperatures and thelower limit at T =
10 MK is obtained with β = .
0, which again coincides with the case of a puresound wave (but with γ = γ = / β between 0.1 and 0.3 can describe the dynamics of the slowMHD wave in the hot loop.We would like to highlight that the correct interpretation of the nature of a slow MHD wave, andhence the density and temperature perturbations observed in the plasma, is essential for the correctinferences of the plasma parameters. We have shown that a finite value of the plasma- β , if not beingsu ffi ciently small, may lead to underestimated results for the adiabatic index γ . In general, coronalactive regions are environments with a very small β leading to experimental values γ e ff being very Article number, page 20 of 23isticò et al.: A failed flare eruption associated with MHD waves
Fig. 10.
Period vs. temperature for a slow wave, assuming phase speed equals the sound speed C S (top) andthe tube speed C T (middle and bottom panels), respectively. close to the real adiabatic index. However, this e ff ect can be pronounced in the chromosphere andin the di ff use or higher corona at a distance greater than 2 R ⊙ , where the approximation β ≪ Article number, page 21 of 23isticò et al.: A failed flare eruption associated with MHD waves
Appendix A: Equations for coronal seismology
General expression for the phase speed and its error for a fundamental standing MHD wave giventhe loop length L ± σ L and the period P ± σ P : V ph = LP , (A.1) σ V ph = V ph p ( σ L / L ) + ( σ P / P ) . (A.2) AppendixA.1: Kinkwaves
The Alfvén speed is determined by the kink speed C K ± σ C K and the densities n e ± σ n e and n ± σ n : C A = C K + n e / n ! / , (A.3) σ C A = | C A | s σ C K C K ! + ( n e / n ) + n e / n ) σ n e n e ! + σ n n ! . (A.4)The magnetic field is given by the loop length, the period and internal and external densitieswith the associated uncertainties: B = LP K q πµ m p n (1 + n e / n ) , (A.5) σ B = | B | s(cid:18) σ L L (cid:19) + (cid:18) σ P P (cid:19) + n + n e ) (cid:16) σ n e + σ n (cid:17) . (A.6) AppendixA.2: Slowwaves
Expression for the sound speed given the plasma temperature T ± σ T : C S = γ k B T µ m p ! / , (A.7) σ C S = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) C S σ T T (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (A.8)The Alfvén speed is given by the tube ( C T ) and sound speeds: C A = C S C T q C − C , (A.9) σ C A = | C A | vt σ C S C S ! + σ C T C T ! + σ C − C C − C , (A.10) ≈ | C A | s σ C S C S ! + σ C T C T ! . (A.11)We note that in our analysis, the quantity ( C − C ) − ≈ − km − s , therefore it can be neglected. Article number, page 22 of 23isticò et al.: A failed flare eruption associated with MHD waves
The internal density n is given by the magnetic field B ± σ B and the Alfvén speed C A ± σ C A : n = (4 πµ m p ) − BC A ! , (A.12) σ n = | n | s σ C A C A ! + (cid:18) σ B B (cid:19) . (A.13) Acknowledgements.
The present work was funded by STFC consolidated grant ST / L000733 / References