Propagating Waves Transverse to the Magnetic Field in a Solar Prominence
B. Schmieder, T.A. Kucera, K. Knizhnik, M. Luna, A. Lopez-Ariste, D. Toot
aa r X i v : . [ a s t r o - ph . S R ] S e p Draft version March 4, 2018
Preprint typeset using L A TEX style emulateapj v. 5/2/11
PROPAGATING WAVES TRANSVERSE TO THE MAGNETIC FIELD IN A SOLAR PROMINENCE
B. Schmieder , T.A. Kucera , K. Knizhnik , M. Luna , A. Lopez-Ariste D.Toot Draft version March 4, 2018
ABSTRACTWe report an unusual set of observations of waves in a large prominence pillar which consist ofpulses propagating perpendicular to the prominence magnetic field. We observe a huge quiescentprominence with the Solar Dynamics Observatory (SDO) Atmospheric Imaging Assembly (AIA) inEUV on 2012 October 10 and only a part of it, the pillar, which is a foot or barb of the prominence,with the Hinode Solar Optical Telescope (SOT) (in Ca II and H α lines), Sac Peak (in H α , H β andNa-D lines), THEMIS (“T´elescope H´eliographique pour l’ Etude du Magn´etisme et des Instabilit´esSolaires”) with the MTR (MulTi-Raies) spectropolarimeter (in He D line). The THEMIS/MTRdata indicates that the magnetic field in the pillar is essentially horizontal and the observations inthe optical domain show a large number of horizontally aligned features on a much smaller scale thanthe pillar as a whole. The data is consistent with a model of cool prominence plasma trapped in thedips of horizontal field lines. The SOT and Sac Peak data over the 4 hour observing period showvertical oscillations appearing as wave pulses. These pulses, which include a Doppler signature, movevertically, perpendicular to the field direction, along thin quasi-vertical columns in the much broaderpillar. The pulses have a velocity of propagation of about 10 km s − , a period about 300 sec, anda wavelength around 2000 km. We interpret these waves in terms of fast magneto-sonic waves anddiscuss possible wave drivers. Subject headings:
Sun: filaments, prominences – Sun: transverse wave – Sun: surface magnetism –Sun: seismology INTRODUCTION
In this work we have an unusually complete set of ob-servations with which to analyze oscillations in a largeprominence footpoint or barb. These allow us to combineinformation concerning both three dimensional motionsand magnetic field to analyze these as fast magnetosonicwaves moving transverse the prominence magnetic fieldsand disturbing the cool material collected in the mag-netic dips.Periodic motions in solar prominences have been rou-tinely observed since the first observations of such oscil-lations in the 1950s (e.g., Ramsey & Smith 1965). Theseoscillations are classified as large- or small-amplitudeoscillations depending on the velocity of the periodicmotions, faster or slower than 20 km s − respectively.Large-amplitude oscillations are related to flare activ-ity which causes almost the entire filament to oscil-late, shaken by the energetic event. Studies of small-amplitude oscillations have revealed a wide range of pe-riods ranging from seconds to hours (Oliver & Ballester2002; Arregui et al. 2012). We will concentrate onthe literature concerning the small amplitude motionswith periods from a few minutes to 10 minutes, calledshort-period oscillations (see the review of Mackay et al.(2010)), that is relevant to our present work. Observatoire de Paris, LESIA, UMR 8109 (CNRS), 92195Meudon, France Code 671, NASA’s GSFC, Greenbelt, MD, 20771 USA Johns Hopkins University, Baltimore, MD USA Instituto de Astrof´ısica de Canarias, E-38200 La Laguna,Tenerife, Spain Universidad de La Laguna, Dept. Astrof´ısica, E-38206 LaLaguna, Tenerife, Spain THEMIS, CNRS-UPS853, E38205 LaLaguna, Spain Alfred University, Alfred, NY, USA
Prominences at the limb exhibit such short period os-cillations (Oliver 1999). Tsubaki & Takeuchi (1986) andTsubaki et al. (1987) reported short period oscillationsin Dopplershift observations of limb prominences. Two-dimensional observations of filaments on the disk haverevealed oscillations transverse to the direction of finestructures of the filaments with a period around 200 sec(Thompson & Schmieder 1991; Yi & Engvold 1991). Re-cent high spatial resolution observations obtained withthe SST show the existence of traveling waves with peri-ods of 3 to 9 minutes in fibril-like structures of filaments.The propagation of the waves are in the same directionas the direction of the prominence fine structures. as themass flows. Okamoto et al. (2007) found that threads ina solar prominence observed with Hinode/SOT under-went vertically oscillating motions with a period around250 sec. Some threads oscillate all along the filamentlength. With Hinode/SOT data, Ning et al. (2009) an-alyzed the oscillations of a quiescent prominence andfound a large variety of oscillations both vertical and hor-izontal with periods of 3 to 6 minutes.The trigger of the small-amplitude, short-period os-cillations has not been detected so far. Many authorsclaimed that the excitation is related with the 3- and5-minutes photospheric and chromospheric oscillations.Solar p -modes are generated by turbulent convection be-neath the photosphere leaking part of their power tothe atmospheric layers: photosphere, chromosphere, andcorona (3- and 5-minutes oscillation).Oscillations in prominences are usually interpreted asan external agent exciting periodic motions on the coolplasma. MHD waves can be propagating or standing.Propagating waves consist of periodic disturbances thatpropagate through the prominence plasma. In contrast,standing waves are confined in a region of the prominencebecause they are the normal modes of the system. Sev-eral theoretical models of prominence oscillations havebeen proposed. Joarder & Roberts (1992a,b) consideredthe whole filament as a vertical plasma slab, ignoring thethread fine structure, with an horizontal straight mag-netic field. The influence of the gravity was considerednegligible and the slab and corona plasmas are uniform.In this model the waves are considered trapped alongthe horizontal direction, and are allowed to propagate inthe vertical direction. Oliver et al. (1992, 1993) includedgravity in the slab model which results in the curvedfield lines of the Kippenhahn-Schl¨uter equilibrium model.The authors found that the gravity introduces only asmall shift on the normal mode frequencies and thus isnot relevant in this kind of configuration. Prominencefine structure consists of threads that could influencethe global prominence oscillations. Joarder et al. (1997)studied theoretically the oscillatory spectra of an isolatedprominence thread. Later, D´ıaz & Roberts (2006) in-cluded this fine-structure and studied fast MHD modesof a periodic, Cartesian multi-threaded model. The au-thors found that the modes behave as propagating modesof an homogeneous prominence with small-scale detailsdue the fibrils.In this paper we report the observations of oscilla-tions in a quiescent prominence observed on the limb ob-tained by several instruments on the ground (THEMIS,Sac Peak) and in space (Hinode/SOT). We describe thecampaign and instrumentation in Section 2. In Section 3we discuss the observations and present the results con-cerning the horizontal magnetic field measured in theprominence and the transverse oscillations. In Section4 we discussed a possible model to explain the observedoscillations. CAMPAIGN AND INSTRUMENTS
The observations were taken during an internationalcampaign organized around Hinode Observation Plan(HOP) 219. Hinode/SOT observed in H α and in Ca IIH lines between 14:04 UT and 18:09 UT. Over a similartime period (14:17-19:37 UT) the Dunn Solar Telescope(DST) at Sacramento Peak Observatory (Sac Peak) wasused to obtain spectra in the H α , H β and Na-D lines.THEMIS (“T´elescope H´eliographique pour l’ Etude duMagn´etisme et des Instabilit´es Solaires” in Canary Is-lands) with the MTR (MulTi Raies) spectropolarimeterobserved the prominence during all the day, obtainingfour full data sets. The best data were collected from10:44-15:30 UT. SDO/AIA with the filters of 304 ˚A and193˚A and STEREO-A EUVI at 195 ˚A are used to supplycontext for the observations. THEMIS
The THEMIS/MTR instrument (L´opez Ariste et al.2000) was used to do spectropolarimetry of the He D line in the observed prominences. The spectrograph slitwas oriented parallel to the local limb. This directiondefined subsequently the sign of the linear polarization:positive Stokes Q means parallel to the slit and, in conse-quence, parallel to the local limb. The double-beam po-larimetry we performed required the use of a grid maskthat presented us with three segments 15.5 ′′ wide alongthe slit, but masked regions of 17 ′′ between each slit. The masked regions allowed us to obtain a double image withopposite polarization, but it also meant that in order toget a continuous covering of the prominence along theslit we had to scan also in this direction by one step of15 ′′ . This extraordinary scan along the slit is the rea-son for the jumps and dark lines in the data from thisinstrument presented below. These artifacts are, how-ever, greatly compensated for by the high quality of thepolarimetry produced by this observing mode.In addition to that scan along the slit a more usualscan perpendicular to the slit was made with steps of 2 ′′ from the limb to the top of the prominence. Altogethertypical fields of view of 120 ′′ × ′′ were covered in aboutone hour with single exposure times of 2 seconds perStokes parameter and scan position. Full polarimetrywith beam-exchange was done with a modulation cycleof 6 images, spanning the three Stokes parameters witheither positive or negative sign measured in every beam,and the simultaneous double beam measuring the oppo-site sign. Each Stokes parameter is thus measured inthe same camera pixel at two different times and in twodifferent pixels at the same time. This symmetry of mea-surements results in a reduction in the systematic errorsto a fourth order perturbation of the signal and high qual-ity measurements. Each cycle was repeated five times toincrease S/N ratios. Hinode/SOT H α and Ca II The Hinode (Kosugi et al. 2007) SOT (Tsuneta et al.2008; Suematsu et al. 2008) consists of a 50-cmdiffraction-limited Gregorian telescope and a Focal PlanePackage including the narrowband filtergraph (NFI),the broadband filtergraph (BFI), the Stokes Spectro-Polarimeter, and Correlation Tracker (CT). For thisstudy, images were taken with a 30 sec cadence in boththe Ca II H line at 3968.5 ˚A using the BFI and at linecenter in the H α line at 6562.8 ˚A using the NFI. TheCa II images have a pixel size of 0.109 ′′ , with a field ofview of 112 × ′′ , while the H α images have a pixel sizeof 0.16 ′′ and a field of view 164 × ′′ . Sac Peak Dunn Solar Telescope
At the DST at Sac Peak we used the Universal Bire-fringent Filter (UBF) to observe filtergrams of H α at linecenter and line center ± . ± . β at line cen-ter and line center ± . − .
25 ˚A. The field of view is about 173 × ′′ .In our analysis we focused on the H α data. a full scan ofthe 5 H α line positions took 10 seconds and time betweenimages at H α line center was 22 to 25.6 sec. Pixel size is0.17 ′′ , with resolution during our observations near 1 ′′ .The full reconstructions of the H α , H β profiles havenot been done yet because it would require us to crosscorrelate the non-simultaneous images obtained in the3-5 points in the profiles and then fit with a Gaussian.This would allow to us to get quantitative estimates ofthe Dopplershifts. It is out of the scope of the presentpaper. We use only movies obtained by computing thedifference of intensities of two symmetric points in theprofiles, a proxy for the Dopplershifts. OBSERVATIONS AND DATA ANALYSIS
Our observations of the prominence oscillations weredone from 14:00 UT -18:00 UT on 2012 Oct 10. Theprominence was observed on the western limb of the Sunat a position angle of about 256 ◦ .The prominence was observed as a filament a few daysbefore the main observations. On October 6 only twoportions of the H α filament are visible in a channel be-tween two active regions (Figure 1). The filament is ori-ented North-South along a meridian. The prominence iswell observed on October 9 and 10 in SDO/AIA filters.As seen in AIA 304 ˚A it consists of a central pillar witharcades on both sides from which material is flowing hor-izontally, mostly outwards away from the central pillarin the plane of the sky (Figure 2). On October 9 thecentral pillar is already observed in absorption as a darkregion over the limb in AIA/193 ˚A. On October 10 atthe time of Hinode observations bright loops are in frontof it and mask the dark area in 193 ˚A. By this time theprominence was already substantially behind the limb asis shown in the image from STEREO-A’s EUVI imagerin Figure 3. Figure 1. H α Filament from BBSO on 2012 October 06 betweenthe active regions NOAA 11582 and NOAA 11584.
Hinode and the ground based instruments (Sac Peakand THEMIS) have smaller fields of view and observedmainly the central pillar (Figure 4).
Wave Observations
Hinode/SOT observed the prominence from 14:04-18:09 UT. The field of view of SOT is centered on the cen-tral foot of the prominence. We see mainly a large, broadpillar with bright, relatively narrow ( < ′′ ) columns con-sisting of horizontal features and lateral extension at thetop. For our measurements here we analyzed the Ca II data, although the waves were apparent in the H α dataas well. The Hinode Ca II data were reduced using thestandard SOT reduction software (fg-prep) in SolarSoft.The SOT/CT (Shimizu et al. 2008) allows fixed track-ing of regions on the disk, but for regions on the limbthere is a slow drift of the target through the field ofview. We correct for this drift using manual correctionsfor large jumps associated to re-pointings and a cross Figure 2. (a) AIA 304 ˚A prominence observed on 2012 Octo-ber 10. The box represents the field of view of Hinode and SacPeak (b) AIA 193 ˚A prominence on 2012 October 09. The box isapproximately the field of view of Hinode/SOT. correlation targeted at the limb and the side of the largescale prominence foot-point. The resulting image seriesis sufficiently stable that remaining jitter and drift do notaffect the motions and changes measured for this paper.The Hinode SOT images show many features whichappear to be wave pulses traveling roughly perpendicu-lar to the solar limb in the narrow columns. The analysisof the most clear of these motions is shown in Figures 4,5, and 6. We integrated the intensity across a 0.55 ′′ (5-pixel) wide area positioned across the oscillating regionas shown in Figure 4 at three different locations, withslits labeled (a), (b), and (c). A 50 minute long intensityslice is shown in Figure 5 (top panel a), corresponding tothe red slit labeled (a). This wave occurs between 1000-2000 seconds, where all times are measured with respectto 14:04:47 UT. In order to remove long term variations,the intensity data were fitted with a quadratic functionwhich was then subtracted from the intensities. The dis-tance between each intensity peak, measured at a giventime, is a measure of the wavelength of the oscillation,and is approximately 2000 km. The slope of the intensitypeaks in Fig. 5 (top panel a), corresponding to the up-wards velocity of the moving features, is approximately10 ± − . The velocity corresponds to the phasespeed of the wave. Since the phase speed remains ap-proximately constant, the wave can be considered to benon-dispersive. In Figure 5 (bottom panel a). we plotthe intensity as a function of time at an altitude of 14 ′′ -900 -800 -700 -600 -500 -400 -300 -200X (arcsec)-600-400-2000 Y ( a r csec ) Figure 3. (a) STEREO/SECCHI/EUVI-A image of the promi-nence observed in the 195 ˚A band at 2012 Oct. 10 17:00:30 UT.The pink curve shows the limb of the Sun as seen from Earth andthus also SDO and Hinode. At this time STEREO-A was at aseparation of 126.18 ◦ ahead of Earth in its orbit. above the photosphere. Four peaks and troughs are seenin this time range. A Fourier analysis of this sectionof the intensity cut, shown in Figure 6 (top panel a),gives a wave period of 277 ±
50 sec. The wavelet analysis,shown in Figure 6 (bottom panel a), indicates that theperiod of the wave remains approximately constant forthe duration of the wave.Identical analyses were performed at two other differ-ent locations, as shown in Figs. 4, 5 and 6 (panels b andc). The wave measured by the intensity slice shown inorange and panel b in Figs. 5 and 6, shows a similar se-ries of bright moving features between 900 and 2000 sec.The speed of the wave is measured to be approximately5 ± − radially outward, and the wavelength isestimated to be roughly 900 km. The intensity cut at analtitude of 5 ′′ above the solar limb is shown in Figure 5(bottom panel b). The period of this wave is seen fromFigure 6 (top panel b) to be consistently near 205 ± ± − and a period of 314 ± α and H β lines obtainedfrom the Sac Peak data show the existence of the sametraveling waves at the same locations as the Hinode Ca II intensity observations. These waves can be detected bothin intensity and in Dopplershift.Figure 7 shows a snapshot of the Sac Peak observations of the prominence (H β Dopplershift ) on Oct. 10 2012.The maxima of the Dopplershifts moving up in the col-umn of the wave are redshifted compared to the wholeprominence.
Magnetic field vector
The raw data of the THEMIS/MTR mode was reducedwith the DeepStokes procedure (L´opez Ariste et al.2009). Data reduction included flat-fielding, dark currentand bias subtraction, wavelength calibration and, partic-ularly, a careful handling of the polarization signals. Theresult of the data reduction are cubes of spectra of the HeD in intensity, linear polarization (both Q and U ) andcircular polarization, for all points along the slit and allpositions of the double scan. S/N ratios are better than10 at the core of the He D in the central parts of theprominence for all three Stokes parameters. With theseS/N ratios we get clear signals of linear polarization asexpected (these are a function of height above the limb,but linear polarization is expected at the level of 10 − times the intensity and therefore 10 times the noise level)but circular polarization is seldom seen above the noise.Whether this circular polarization is due to Zeeman ef-fect or the alignment-to-orientation transfer mechanism(L´opez Ariste & Casini 2002) those low signals alreadypoint to weak magnetic fields, a conclusion that will beconfirmed by the inversion codes.The Stokes profiles are fed to an inversioncode based on Principal Component Analysis(L´opez Ariste & Casini 2002; Casini et al. 2003) thatefficiently compares the observed profile against thosein a database generated with known models of thepolarization profiles of the He D . The comparison ismade pixel by pixel independently. The database usedcontains 90000 profiles computed as the emission of asingle He atom in its triplet state modeled with the 5levels of lower energy of the He triplet system. The atomis polarized by the anisotropic radiation of the photo-sphere below the prominence at different heights (oneof the free parameters of the model). Collisions are nottaken into account. The atomic polarization of the Heatom is modified by a single vector magnetic field withfree strength, inclination and azimuth. The Hamiltonianof the atom includes all terms with its Zeeman sublevelssplitting linearly with the magnetic field. We solve thedensity matrix of the atom in statistical equilibrium;the solution contains all populations and quantumcoherence, including atomic alignment and orientation,for all the levels involved in the He triplet atom model.The Hanle effect of every level is thus computed as wellas the Zeeman effect. From the resulting populationsand coherence we compute the polarization dependentemission terms, in whatever direction we are observing.The scattering angle is thus a free parameter of themodel too. Several million profiles thus computed areused to fill the database, keeping just those which aredifferent enough among them and rejecting others sothat the database fills as homogeneously as possible thespace of possible profiles while keeping at a small size.After comparison of any observed profile with those inthe database, the most similar is kept as the solution andthe parameters of the model used in its computation arekept as the inferred vector magnetic field, height above HINODE SOT/WB 10-Oct-2012 14:04:47.098 UT
910 920 930 940 950X (arcsecs)-350-345-340-335-330-325-320 Y ( a r cs e cs ) (a)(b)(c) Figure 4. left panel:
Hinode prominence in the Ca II H line at 14:00 UT on October 10 2012. The box is approximately the field of viewof THEMIS/MTR. right panel:
Hinode/SOT image showing the location of the three 5 pixel, 0.55 ′′ wide slits along which the intensity ofthe oscillation shown in Fig. 5 was calculated. The labels (a), (b), and (c) correspond to panels in Figs. 5 and 6. the photosphere and scattering angle. Error bars aredetermined for those parameters as well by doing somestatistics on all other models which are sufficiently sim-ilar to the observed ones, though not as similar as theone selected as solution. It is important to stress thatalthough there is always one case in the database that isthe most similar one to the observed one, this does notmean that it is a good fit to all of the observed profiles. Itis thus important to keep a measure of how similar theyare and also to check that all conclusions on the mag-netic field strength or orientation are based upon sets ofprofiles that really correctly represent the observation.Figure 8 presents three of the four maps of theprominence intensity obtained on October 10 withTHEMIS/MTR in the He D line.Figure 9 presents the maps obtained after inversionof the Stokes parameters recorded in the He D linewith THEMIS/MTR : (a) Intensity, (b) Magnetic fieldstrength (c) Inclination, (d) Azimuth. The angle originof inclination is the local vertical, the origin of the az-imuth is a plan containing the line of sight and the localvertical. We see that the brightest parts of the promi-nence have an inclination of 90 ◦ which means that themagnetic field in these bright columns is horizontal.Figure 10 presents the variation of magnetic fieldstrength, inclination and azimuth along the brightestcolumn in Figure 9. The field strength in the brightcolumn is around 7.5 Gauss and horizontal. The az-imuth is close to 100 ◦ . This means that the magneticfield vector is mainly in the plane of the sky. This con-firms previous results (Bommier & Leroy 1998). Obser-vations of prominence footpoints observed on the diskhave also shown that the field lines are tangent to thephotosphere (L´opez Ariste et al. 2006). Linear force freefield extrapolations show that prominence plasma is sup-ported by shallow dips in magnetic field lines like inthe Kippenhahn-Schl¨uter model (Aulanier & D´emoulin1998; Dud´ık et al. 2008). The feet or footpoint of aprominence would be piles of dips in horizontal field lines.The existence of a dip in magnetic field lines to represent a prominence thread has been discussed accordingly tothe value of the β plasma by Heinzel & Anzer (1999).Our observation of prominence fits closely the side viewof a footpoint of prominence (or barb) modeled by dipsof magnetic field lines in Figure 5 of Dud´ık et al. (2012).The global shape of the foot is like an “anvil” and eachportion of the field lines is horizontal and roughly in theplane of the sky. FAST MAGNETOSONIC WAVE MODEL
Theoretical Phase Velocity of Waves
Propagating waves have been detected in three verticalbright columns observed by Hinode/SOT and the DST.In the column with the brightest Ca II intensity (ana-lyzed with slit a, Fig. 4) the observed wave train propa-gates upwards in the prominence with a projected phasespeed of around 10 km s − , a projected wavelength of2000 km, and a period of 277 s as described in Sec. 3.1.These intensity variations in the Ca II filters are inter-preted in terms of plasma compressions and rarefactions.According to the magnetic field measurements the prop-agation of the waves are mainly perpendicular to thehorizontal local magnetic field. Prominences have a com-plex fine structure that consists of thin threads (Lin et al.2005) that extend along the magnetic field. The brightcolumns consist of many piled-up threads. The wavepropagates in a highly non-homogeneous medium; how-ever, the measured phase speed seems independent of theposition of the wavefront (see Section 3.2).As a first approximation we consider that waves propa-gate in a uniform medium, and we interpret the observedwaves as fast magnetosonic modes. These wave modespropagate perpendicularly to the local magnetic field andthe velocity perturbations are longitudinal, along thepropagation direction. The restoring force of these wavesare the magnetic and gas pressure, and it produces com-pressions and rarefactions of the plasma and the mag-netic field intensity. Based on the observed properties,we explore the different wave modes that could be rele-vant to the situation by doing a theoretical phase velocity D i s t an c e ( a r cs e cs ) a) D i s t an c e ( a r cs e cs ) b) D i s t an c e ( a r cs e cs ) c) I n t en s i t y a) I n t en s i t y b) I n t en s i t y c) Figure 5. top panels (a) (b) (c) Intensity maps as a function oftime along the three slits shown in Figure 4. Cuts of the intensitywas made along the horizontal line, shown in bottom panels (a) (b)(c ). The vertical lines represent the range of times of the dataused for the periodogram results in Fig. 6. Long term trends weresubtracted from the intensities resulting in negative values. diagram. In Figure 11 we have plotted the sound speed, c sound , the Alfv´en speed, v Alfv´en , and the magnetosonicspeed given by v ms = q v + c , (1)as a function of the electron number density, n e andthe averaged magnetic field intensity of our observa-tions of 7 . n e ∼ − cm − (Labrosse et al. 2010) the plasma- β is smalland v Alfv´en > c sound . Our observations reveal a projectedspeed of the order of 10 km s − that is much smallerthan the magnetosonic speeds. In fact, it is similar to c sound corresponding to the slow modes. However, theslow modes propagates mainly along the magnetic fieldin the range of small β , that is perpendicular to the ob-served direction. The discrepancy between the observedand theoretical values could be explained by the projec-tion effect. The waves form an angle with respect to the
100 200 300 400 500Period (sec)0.02.24.46.68.811.0 P o w e r a)
100 200 300 400 500Period (sec) b)
100 200 300 400 500Period (sec) c)
359 796 1233 1670 2107 2545Time (sec)1000720518373268193139100 P e r i od ( s e c ) % % % a) % % b) % % c) Figure 6. top panels (a) (b) (c) Periodograms of the intensity cutshown in fig. 5 The peak power is located respectively at 277 sec( ±
50 sec), 205 sec ( ±
54 sec) and 314 sec ( ±
125 sec). bottom panels (a) (b) (c) Confidence regime for the periodograms, indicating thatthe period is staying approximately constant for the first two wavesbut many be changing for the third wave.
LOS, α LOS . In the same Figure 11 we have plotted theangle α LOS necessary to have a projected v ms velocity of10 kms − defined as α LOS = arcsin(10 /v ms ). This indi-cates that for the range of typical prominence densitiesthe wave moves in a direction forming an small angle withrespect to the LOS of α LOS < ◦ (or α LOS > ◦ ).These angles are very extreme indicating that for typicalprominence values the propagation is mainly along theLOS. Another possibility is that the prominence electrondensity is larger than 10 cm − and the projection angleis small. For example with n e = 5 × cm − the mag-netosonic speed is of around 22 kms − and α LOS = 27 ◦ .Thus, a combination of a relatively small α LOS and rel-atively large prominence density can explain the smallphase speed of the observed magnetosonic waves.
Model of Waves
The propagating waves seem to be confined in the coolprominence column. This could be because we only ob-serve the cool plasma in our observations, and the vis-ibility of the waves are difficult to observe outside thecool prominence. However, theoretical models predictthat the waves are truly confined in the prominence,
Figure 7.
Dopplershifts in the prominence derived from Sac PeakH β observations (173 x173 ′′ ). Note the large Dopplershift in thecolumn (similar to the column indicated by a red slit in the Fig.4right panel) pointed by an arrow, the box is approximately the fieldof view of Hinode/SOT. as described by Joarder & Roberts (1992a,b, 1993) andOliver et al. (1992, 1993). These authors modeled theprominence as a slab of cool plasma using different con-figurations of the magnetic field. These systems exhibitmany normal modes that are confined or trapped inthe horizontal direction but propagate vertically. Theseworks concentrated on the large wavelength limit, k z L ≪
1, where k z is the vertical wavenumber and L is thelength of the prominence field lines. In that situationthe whole prominence oscillates in phase in the verticaldirection. However, in the waves described in this workthe wavelength is short compared with the length of theprominence magnetic field so that k z L ≫
1. A detailednormal mode analysis of the system is out of the scopeof this paper; it will be a subject of a future work.For the present work, we have modeled the prominenceas an uniform plasma slab with a uniform horizontalmagnetic field (see Figs. 12a and 12b). The average mag-netic field intensity is set to 7.5 Gauss. We have madea time dependent simulation of the system described byJoarder & Roberts (1992b). The model consists of a ver-tical prominence slab with a horizontal magnetic fieldtransverse to the prominence slab. Gravity is neglectedbecause it has little influence on the waves (Oliver et al.1992). In this simulation we try to understand how thewaves propagate in the prominence structure and nothow the waves are produced and reach the prominence.For this reason we have considered the prominence slabwith an infinite vertical extension and we have not mod-eled the photosphere, chromosphere, and transition re-gion. The model atmosphere is uniform, the top of thephotosphere is placed at z = 0. The driver perturbs the v z -component of the velocity and consists in a planarpulse located at z = −
10 Mm with a Gaussian shape,
Figure 8.
THEMIS/MTR observations of the prominence in HeD line intensity: between (a) 10:44 and 11:52 UT, (b) 12:09 and13:12 UT, and (c) 14:26 and 15:30 UT. The fields of view are 42 ′′ × ′′ . The fields of view are not the same. They have been shiftedby a few pixels towards the top of the prominence. The images arerotated so that the limb is horizontal. The dark vertical lines aredue to the grid mode of the observations. v z = e − ( z +103 ) . The width of the Gaussian is 3 Mm.We have checked several driver shapes and all producesimilar results. The driver oscillates three times witha period of 300 s that is similar to the observed one.The prominence-slab is placed between x = − x = 5 Mm, with a total width of 10 Mm. The electrondensity of the prominence is set to n e = 5 × cm − ,100 times larger than that of the surrounding corona.The temperature of the prominence plasma is 8000 Kand the corona is 10 K. We fulfill the pressure imbal-ance condition ρ p T p = ρ c T c , where the p and c subscriptsrefer to prominence and coronal medium. The equilib- Figure 9.
THEMIS/MTR observations of the prominence in HeD line : (a) Intensity between 10:44 and 11:52 UT (b) Magneticfield strength (c) Inclination, (d) Azimuth. The color chart refersfrom 0 ◦ to +180 ◦ (left to right) . Orange means around 90 ◦ . Theinclination is measured from the vertical. All the orange pixels inthe inclination map (on the left ) show that the field direction ismainly horizontal. The azimuth is mainly around 110 ◦ , so that thefield direction is directed approximately (within about 30 ◦ ) parallelto the plane of the sky rium magnetic field is uniform and horizontal, ~B = B ˆ x ,with B = 7 . . − , which coincides withthe speed of the magnetosonic waves as we see in Figure11. We have performed several numerical experimentswith different configurations and in all the cases the prop-agation speed coincides with the magnetosonic velocity.In Joarder & Roberts (1992b, 1993), the authors foundthat the short period oscillations are essentially trappedmagnetosonic waves reflecting off the boundaries of theprominence slab and propagating along it. The normalmode excited in the simulations can be identified by theso-called string mode fIF. This mode is classified as inter-nal mode by Joarder & Roberts (1992b) where the veloc-ity of the perturbation is mainly located inside the promi-nence. Oliver et al. (1993) classified more accurately thenormal modes of a prominence and found that the fIFmodes are hybrid. This means that removing the coro-nal or prominence medium the mode remains, but witha different frequency and shape.Our simulations demonstrate that the vertically propa-gating waves transverse to the magnetic field are magne-tosonic waves ducted along the prominence column andthe treatment of waves in an homogeneous medium isapplicable. The observed prominence is quiescent andseems quite static in the observed time. This indicatesthat the plasma β is small and then v Alfv´en > c sound andthe observed waves are fast magnetosonic modes. In or-der to have a projected magnetosonic speed comparableto the observed velocity the projection angle α LOS shouldbe small. In fact, assuming that the prominence densityhave a typical n e , α LOS < ◦ . As the wave is ductedin the prominence this indicates that the column of coolplasma forms and small angle with respect to the LOS.This could be associated to that the column is not placedexactly at the solar limb at the moment of the observa-tions. In Figure 3 we see that the dark pillar is consid-erably behind the limb. It is almost 150 arcsecs behind,that is almost 10 ◦ with respect to the Sun’s center. Thismeans that the local vertical to the solar surface at thepillar position is 80 ◦ instead of 90 ◦ with respect to theLOS. Additionally the bright column forms part of theprominence foot or barb, and such structures are oftenvery elongated when seen on the disk but are not very Figure 10.
From left to right, plots of the inferred magnetic field strength (G), inclination (degrees, 90 ◦ being horizontal) and azimuth(degrees, 90 ◦ being in the plane of the sky). They are plotted along a cut through the brightest prominence column (similar to the columnindicated by slit (a) in the Fig.4 right panel). The diamonds are the actual data, while the shaded region is the 3-sigma smoothed confidenceregion of the results. The azimuth is subject to a 180 ◦ ambiguity that has been solved ad-hoc but that invalidates the computation of theconfidence limits. Figure 11.
Plot of the theoretical phase velocity of the magne-tosonic mode (solid line), the slow mode (dashed line), the Alfv´enmode (dotted line) as a function of the electron number densityof the prominence plasma. The magnetic field intensity used is7 . n e ∼ − cm − (see, Labrosse et al. 2010). The dark areacorresponds to larger values of n e and we have plotted this val-ues in order to have values of the theoretical magnetosonic speedcomparable to the observed velocities. We have also plotted witha dot-dashed line the α LOS = arcsin(10 /v ms ) defined as the an-gle should form the propagation direction of a magnetosonic wavewith respect to the LOS in order to have a projected velocity of10 km s − (the observed value). tall when seen at the limb, indicating large inclinationsof those structures with respect to the solar surface. InFigure 3 we can see that the horizontal extension of thepillar is of order of 50 arcsecs, and the vertical extensionis less than 50 arcsecs from Figure 4. Then this meansthat the pillar is inclined more than 45 ◦ with respect tothe vertical direction. Numerical simulations indicatesthat prominence foot consist of many horizontal threadpilled to form inclined structures with respect to the so-lar vertical (see Fig. 7 by Dud´ık et al. 2012). In Figure 13 we have included a sketch of the configuration of thepillar in the observation. The prominence foot consistsin a pillar of threads that are placed in the dips of themagnetic field lines. The pillar is behind the limb (redline) as the observations reveal (see Fig. 3). The pillarhas also an inclination with respect to the local vertical.Thus, it is plausible and consistent with our model thatthe oscillating columns form a small angle with respectto the LOS ( ≈ ◦ ) and have a relatively large density(a few times 10 cm − ).The DST observations reveals Doppler velocities (see § α LOS = 90 ◦ the velocity of oscillation has aprojection along the LOS.Along the third slit we analyzed (slit c) we observeddownward motion, and visually it appears that somewaves move upwards and then reflect back downwards.These motions can be attributed to the propagation ofthe waves in an non-homogeneous medium. The studyof the propagation of the waves in such a complicatedmedium will be a subject of a future work. CONCLUSION
A quiescent prominence has been observed with Hin-ode (Ca II H, H α ), Sac Peak (H β and H α ) over a timeperiod of 4 hours, THEMIS/MTR (vector magnetogramsin He D ), SDO/AIA (193 ˚A, 304 ˚A), and STEREO-A/EUVI (195 ˚A) on October 10 2012. The small fieldof view of the later instruments is centered on one largefoot of the prominence. This foot appears in 304 ˚A, as alarge quasi-vertical pillar with material flowing on eachside along horizontal field lines. The polarimetry in theHe D line obtained by the observations of THEMIS inthe MTR mode allows us to derive the magnetic fieldstrength, the inclination and the azimuth in the regionof the prominence observed. The magnetic field is mainlyhorizontal with a field strength around 5 -10 Gauss.The observed waves propagate perpendicularly to the0 Figure 12.
Time-dependent simulation results of a wave traintraveling along the prominence. The prominence slab is placedbetween x = − x = 5 Mm with an uniform electrondensity of 10 cm − . The magnetic field is horizontal and uniform ~B = 7 . x . In (a) the wavefront appears from below and in (b) thefull wave train has appeared traveling upwards. The gas pressure(color) shows almost planar wavefronts confined in the prominencecolumn. Similarly, the velocity field (vectors) is almost verticalinside the prominence with the largest values, whereas is almosthorizontal outside with smaller values. magnetic field according to the measurements of the mag-netic field by THEMIS. The observed phase speeds arebelow 10 km s − , the periods are around 300 s, and thewavelengths are about 2000 km. Our simulations revealsthat fast magnetosonic waves are ducted along the promi-nence foot moving upwards and probably away from theobserver. These waves produce compressions and rar-efactions of the plasma as the observations. The direc-tion of the propagation are perpendicular to the magneticfield as expected. We conclude that the observed wavesare fast magnetosonic waves ducted in the prominencefoot. We can explain the small projected observed phasespeed as a combination of a relatively small α LOS andrelatively large prominence density.In some cases it seems that the wavefronts prop-agates downwards after a reflection at some height.This could be associated with non-homogeneities of theplasma and magnetic field. However, the waves are ob-
Figure 13.
Sketch of the possible configuration of the pillar andthe observation. The pillar consists in threads located at dips thatare pilled up. The pillar direction (dotted line) is inclined withrespect to the local vertical to the solar surface (orange surface).The pillar is placed behind the solar limb (red line). The dashedline is the LOS with the observer placed at the left hand side ofthe plot. The LOS forms an angle α LOS with respect to the pillar.In the situation of a vertical pillar placed at the limb the angle is90 ◦ . The inclination of the pillar and the position behind the limbmake an angle, α LOS which is smaller than 90 ◦ . served emanating from the direction of the solar sur-face. We cannot identify the driver of these waves.We speculate that they could be associated with rapidspicule jets, magnetic reconnection in the photosphereor just above, due to the emerging of small flux closeto the prominence, or that the waves are associatedwith the characteristic 5-minutes photospheric oscil-lations tunneled to the chromosphere (Heggland et al.2011; Khomenko & Calvo Santamaria 2013). We ob-serve waves during the entire 4 hour period of observa-tion. This fact could be important for the coronal heat-ing problem because these waves contributes to the ACheating. In a future work we will investigate the propaga-tion of the waves in non-homogeneous plasmas and howdifferent triggers can excite these waves in prominences.Hinode is a Japanese mission developed and launchedby ISAS/JAXA, collaborating with NAOJ as a domes-tic partner, NASA and STFC (UK) as internationalpartners. Scientific operation of the Hinode missionis conducted by the Hinode science team organized atISAS/JAXA. This team mainly consists of scientistsfrom institutes in the partner countries. Support for thepost-launch operation is provided by JAXA and NAOJ(Japan), STFC (U.K.), NASA, ESA, and NSC (Norway).SDO data are courtesy of NASA/SDO and the AIA sci-ence team. The STEREO/SECCHI data are producedby an international consortium of the NRL, LMSAL andNASA GSFC (USA), RAL and Univ. Bham (UK), MPS(Germany), CSL (Belgium), IOTA and IAS (France).The authors BS and AL would like to thank B. Gellythe director of THEMIS and the technical team to haveallowed to do these observations using the THEMIS in-strument in Canaries Islands. TK would like to ac-knowledge support from the NASA Living with a Starprogram. ML gratefully acknowledge partial financialsupport by the Spanish Ministry of Economy throughprojects AYA2011-24808 and CSD2007-00050. This workcontributes to the deliverables identified in FP7 Euro-pean Research Council grant agreement 277829, ” Mag-netic connectivity through the Solar Partially Ionized At-mosphere”, whose PI is E. Khomenko.1served emanating from the direction of the solar sur-face. We cannot identify the driver of these waves.We speculate that they could be associated with rapidspicule jets, magnetic reconnection in the photosphereor just above, due to the emerging of small flux closeto the prominence, or that the waves are associatedwith the characteristic 5-minutes photospheric oscil-lations tunneled to the chromosphere (Heggland et al.2011; Khomenko & Calvo Santamaria 2013). We ob-serve waves during the entire 4 hour period of observa-tion. This fact could be important for the coronal heat-ing problem because these waves contributes to the ACheating. In a future work we will investigate the propaga-tion of the waves in non-homogeneous plasmas and howdifferent triggers can excite these waves in prominences.Hinode is a Japanese mission developed and launchedby ISAS/JAXA, collaborating with NAOJ as a domes-tic partner, NASA and STFC (UK) as internationalpartners. Scientific operation of the Hinode missionis conducted by the Hinode science team organized atISAS/JAXA. This team mainly consists of scientistsfrom institutes in the partner countries. Support for thepost-launch operation is provided by JAXA and NAOJ(Japan), STFC (U.K.), NASA, ESA, and NSC (Norway).SDO data are courtesy of NASA/SDO and the AIA sci-ence team. The STEREO/SECCHI data are producedby an international consortium of the NRL, LMSAL andNASA GSFC (USA), RAL and Univ. Bham (UK), MPS(Germany), CSL (Belgium), IOTA and IAS (France).The authors BS and AL would like to thank B. Gellythe director of THEMIS and the technical team to haveallowed to do these observations using the THEMIS in-strument in Canaries Islands. TK would like to ac-knowledge support from the NASA Living with a Starprogram. ML gratefully acknowledge partial financialsupport by the Spanish Ministry of Economy throughprojects AYA2011-24808 and CSD2007-00050. This workcontributes to the deliverables identified in FP7 Euro-pean Research Council grant agreement 277829, ” Mag-netic connectivity through the Solar Partially Ionized At-mosphere”, whose PI is E. Khomenko.1