Kink oscillations of flowing threads in solar prominences
aa r X i v : . [ a s t r o - ph . S R ] J un Astronomy&Astrophysicsmanuscript no. 16536 c (cid:13)
ESO 2018September 26, 2018
Kink oscillations of flowing threads in solar prominences
R. Soler , and M. Goossens Centre for Plasma Astrophysics, Department of Mathematics, Katholieke Universiteit Leuven, Celestijnenlaan 200B, 3001 Leuven,Belgiume-mail: [email protected] Solar Physics Group, Departament de F´ısica, Universitat de les Illes Balears, E-07122 Palma de Mallorca, SpainReceived XXX / Accepted XXX
ABSTRACT
Context.
Recent observations by Hinode / SOT show that MHD waves and mass flows are simultaneously present in the fine structureof solar prominences.
Aims.
We investigate standing kink magnetohydrodynamic (MHD) waves in flowing prominence threads from a theoretical point ofview. We model a prominence fine structure as a cylindrical magnetic tube embedded in the solar corona with its ends line-tied inthe photosphere. The magnetic cylinder is composed of a region with dense prominence plasma, which is flowing along the magnetictube, whereas the rest of the flux tube is occupied by coronal plasma.
Methods.
We use the WKB approximation to obtain analytical expressions for the period and the amplitude of the fundamental modeas functions of the flow velocity. In addition, we solve the full problem numerically by means of time-dependent simulations.
Results.
We find that both the period and the amplitude of the standing MHD waves vary in time as the prominence thread flowsalong the magnetic structure. The fundamental kink mode is a good description for the time-dependent evolution of the oscillations,and the analytical expressions in the WKB approximation are in agreement with the full numerical results.
Conclusions.
The presence of flow modifies the period of the oscillations with respect to the static case. However, for realisticflow velocities this e ff ect might fall within the error bars of the observations. The variation of the amplitude due to the flow leads toapparent damping or amplification of the oscillations, which could modify the real rate of attenuation caused by an additional dampingmechanism. Key words.
Sun: filaments, prominences — Sun: oscillations — Sun: corona — Magnetohydrodynamics (MHD) — Waves
1. Introduction
Recent observational evidence of ubiquitous periodically vary-ing features in the solar corona (e.g., Tomczyk et al. 2007;Jess et al. 2009; Tomczyk & McIntosh 2009; Wang et al. 2009)has raised the debate on whether these observations are causedby magnetohydrodynamic (MHD) waves or by quasi-periodicflows (see, e.g., De Pontieu & McIntosh 2010). There seem tobe strong theoretical arguments supporting the wave interpreta-tion (e.g., Erd´elyi & Fedun 2007; Van Doorsselaere et al. 2008;Terradas et al. 2010; Verth et al. 2010). However, waves andflows are not mutually exclusive and, in fact, both phenomenahave been simultaneously observed in the fine structure of solarprominences (e.g., Okamoto et al. 2007). This o ff ers us the op-portunity to study the interaction between waves and flows in thesolar atmosphere.The fine structure of solar prominences is clearly visi-ble in the high-resolution H α and Ca II H-line images fromthe Solar Optical Telescope (SOT) aboard the Hinode satellite(e.g., Okamoto et al. 2007; Berger et al. 2008; Chae et al. 2008;Ning et al. 2009; Schmieder et al. 2010; Chae 2010). When ob-served above the limb, vertical structures are commonly seenin quiescent prominences (e.g., Berger et al. 2008; Chae et al.2008; Chae 2010), while horizontal threadlike structures are usu-ally observed in active region prominences (e.g., Okamoto et al.2007). Although it is apparently di ffi cult to reconcile both pic-tures, some authors (e.g., Schmieder et al. 2010) have suggested Send o ff print requests to : R. Soler that vertical threads might actually be a pile up of horizontalthreads which appear as vertical structures when projected on theplane of the sky. This idea is consistent with H α observations offilaments on the solar disk from the Swedish Solar Telescope(e.g., Lin et al. 2008, 2009), in which the filament fine struc-ture is seen as thin and long dark ribbons. On the other hand,other authors (e.g., Chae 2010) have argued that vertical threadsare real and are an indication of the existence of vertical mag-netic fields in quiescent prominences. Thus, it remains unclearwhether all prominences have the same magnetic structure or,on the contrary, the magnetic field in quiescent prominences ispredominantly vertical and active region prominences have hor-izontal fields. A recent review on the properties of prominencethreads can be found in Lin (2010).There are many evidences of transverse oscillations ofthe fine structures of both active region and quiescent promi-nences, which have been interpreted in terms of kink MHDwaves (see the recent reviews by Ballester 2006; Oliver 2009;Arregui & Ballester 2010). The reported periods are usually ina narrow band between 2 and 10 minutes, while the oscilla-tions are typically damped after a few periods. In addition, flowsand mass motions in prominences have been also reported (e.g.,Zirker et al. 1998; Wang 1999; Kucera et al. 2003; Lin et al.2003; Ahn et al. 2010). The typical flow velocities are less than30 km s − in quiescent prominences, although larger values up to40–50 km s − have been observed in active region prominences.The work of Okamoto et al. (2007) is an example of simul-taneous transverse oscillations and mass flows in prominencefine structures observed with Hinode / SOT. In the present pa-
R. Soler and M. Goossens: Kink oscillations of flowing prominence threads per we focus on the theoretical analysis of the event reportedby Okamoto et al. (2007). Similar observations of simultaneousflows and oscillations have been reported by Ofman & Wang(2008) in coronal loops and by Cao et al. (2010) in filament foot-points. Also, the recent work by Antolin & Verwichte (2011)on observations of transverse oscillations of loops with coro-nal rain is relevant for our present theoretical investigation.Okamoto et al. (2007) observed an active region prominenceformed by a myriad of horizontal magnetic flux tubes whichare partially outlined by threads of cool and dense prominenceplasma. The magnetic tubes are probably rooted in the solarphotosphere. Although only the part of the tubes filled withprominence material can be seen in the Ca II H-line images, thelength of the whole magnetic tube must be much longer thanthe length of the prominence threads, which is roughly between3,000 km and 16,000 km. Okamoto et al. (2007) detected thatsome threads were flowing along the magnetic tubes and simul-taneously oscillating in the vertical direction. The mean periodof the oscillations was 3 min and the apparent flow velocity onthe plane of sky was around 40 km s − . The oscillations were inphase along the whole length of the threads, and the wavelengthwas estimated to be at least 250,000 km.The event observed by Okamoto et al. (2007) was stud-ied from a theoretical point of view by Terradas et al. (2008),who interpreted the observations in terms of standing kinkMHD modes supported by the magnetic structure (see, e.g.,Edwin & Roberts 1983; D´ıaz et al. 2002; Goossens et al. 2009;Soler et al. 2010). An interpretation of the observations byOkamoto et al. (2007) in terms of kink modes was also sug-gested by Erd´elyi & Fedun (2007) and Van Doorsselaere et al.(2008). Terradas et al. (2008) used the observed wave propertiesprovided by Okamoto et al. (2007) to perform a seismologicalestimation of a lower bound of the prominence Alfv´en speed.The time-dependent numerical simulations by Terradas et al.(2008) suggested that the influence of the flow on the periodwas small. Nevertheless, the precise e ff ect of the flow was notassessed in their work because a detailed parametric study wasnot performed. The purpose of this paper is to advance the anal-ysis of the event observed by Okamoto et al. (2007) by com-bining both analytical and numerical methods. In the analyti-cal part, we use the WKB approximation to assess the e ff ectof the flow on the period and the amplitude of the transverseoscillations. Expressions of these quantities as functions of therelevant parameters of the model are obtained. In the numericalpart, we go beyond the WKB approximation and solve the fulltime-dependent problem. The implications of our results for themagneto-seismology of prominences are also discussed.This paper is organized as follows. Section 2 contains themodel configuration and the basic governing equations. The an-alytical investigation of standing kink MHD waves in flowingprominence threads using the WKB approximation is includedin Section 3, while the full numerical solution of the time-dependent problem is performed in Section 4. Finally, our resultsare discussed in Section 5.
2. Model and basic equations
The background model in which the waves are superimposed isschematically shown in Figure 1. It is composed of a straightand cylindrical magnetic flux tube of radius R and length L ,whose ends are fixed at two rigid walls representing line-tyingat the solar photosphere. The z -axis is chosen so that it coincideswith the axis of the tube, and the photospheric walls are locatedat z = ± L /
2. The magnetic tube is partially filled with promi- nence plasma of density ρ p , while the rest of the tube, i.e., theevacuated part, is occupied by less dense plasma of density ρ e .The density of the external plasma is the density of the coronalmedium, ρ c . The length of the prominence region (thread) is L p .The thread flows along the tube as a block with constant speed v . The magnetic field is B = B ˆ e z , with B homogeneous. As the β = β theratio of the gas pressure to the magnetic pressure, the plasmatemperature is irrelevant for the study of kink MHD wavessupported by the model. In the absence of flow, standing kinkMHD waves supported by the present model were investigatedby Joarder et al. (1997) and D´ıaz et al. (2001) in Cartesian ge-ometry, and by D´ıaz et al. (2002), Dymova & Ruderman (2005),D´ıaz et al. (2010), and Soler et al. (2010) in cylindrical geome-try. We adopt the TT approximation, which is valid for R / L ≪ R / L p ≪
1. To check whether or not this approximation isreasonable in the context of prominence threads, we take intoaccount that the values of R and L p reported by the observations(e.g., Lin 2004; Okamoto et al. 2007; Lin et al. 2008) are in theranges 50 km . R .
300 km and 3,000 km . L p . L ∼ km as a typical length forthe magnetic tube. We obtain R / L p and R / L in the ranges3 × − . R / L p . . × − . R / L . × − ,meaning that the use of the TT approximation is justified inprominence fine structures. In the case v =
0, the basic equa-tion governing linear kink MHD waves of the flux tube in theTT approximation was derived by Dymova & Ruderman (2005)in their Equation (21). In the absence of flow, the TT approx-imation was used by Dymova & Ruderman (2005), D´ıaz et al.(2010), and Soler et al. (2010). The results of these works fullyagree with the general results beyond the TT approximation byJoarder et al. (1997), D´ıaz et al. (2001), and D´ıaz et al. (2002).In the presence of flow, an intuitive generalization ofEquation (21) of Dymova & Ruderman (2005) was performedby Terradas et al. (2008) in their Equation (2). Mathematically,Morton & Erd´elyi (2010a) also considered the variation of den-sity with time and obtained a similar expression in theirEquation (18). We refer the reader to Terradas et al. (2008) andMorton & Erd´elyi (2010a) for a detailed derivation of the basicequation. In the mathematical derivation of Morton & Erd´elyi(2010a) it is assumed that the di ff erence of the flow velocitybetween the internal and external plasma is small, i.e., muchsmaller than the Alfv´en velocity. So, we restrict our present in-vestigation to values of the flow velocity that satisfy v / v Ap ≪ v Ap = B √ µρ p is the prominence Alfv´en speed. Assuming B =
50 G and ρ p = − kg m − as realistic values of the mag-netic field strength and density in active region prominences, weobtain v Ap ≈
446 km s − . Since the flow velocities on the planeof sky estimated by Okamoto et al. (2007) are in the interval be-tween 15 km − to 46 km s − (see their Table 1), the restriction v / v Ap ≪ ∂ v r ( z , t ) ∂ t − v ( z , t ) ∂ v r ( z , t ) ∂ z = , (1)which has to be solved along with the condition of line-tying atthe photosphere expressed as v r ( ± L / , t ) =
0, and a given initialcondition at t =
0. In Equation (1), v r ( z , t ) is the radial velocityperturbation at the tube boundary and v k ( z , t ) is the kink speed,which in our model is a function of z and t , namely v k ( z , t ) = ( v kp if | z − z − v t | ≤ L p / , v ke if | z − z − v t | > L p / , (2) . Soler and M. Goossens: Kink oscillations of flowing prominence threads 3 Fig. 1.
Sketch of the prominence fine structure model adopted in this work.where v kp = vt B µ (cid:16) ρ p + ρ c (cid:17) , v ke = s B µ ( ρ e + ρ c ) , (3)with µ the magnetic permittivity, and z corresponds to the po-sition of the center of the prominence thread with respect tothe center of the magnetic tube at t =
0. Note that z < z > v k ( z , t ) given in Equation (2). Terradas et al. (2008) per-formed time-dependent simulations and solved Equation (1) nu-merically. Here, our aim is to solve Equation (1) by using bothanalytical and numerical methods.
3. Analytical investigation: WKB approximation
We solve Equation (1) analytically by using the Wentzel-Kramers-Brillouin (WKB) approximation (see, e.g.,Bender & Orszag 1978, for details about the method). TheWKB approximation has been recently applied to the investi-gation of MHD waves in cooling coronal loops (Morton et al.2010; Morton & Erd´elyi 2010a,b). In particular, the work byMorton & Erd´elyi (2010a) is especially relevant for the presentinvestigation as they studied kink oscillations of coronal loopswith variable background.The WKB approximation is an approximate method to studywaves in a changing background whose properties are smoothfunctions of space and / or time. In the present application ofthe WKB approximation we assume that the time scale relatedto the waves, e.g., the period, is much shorter than the timescale related to the changes of the background configuration.Under these conditions, it is possible to define a time-dependentfrequency which slowly varies because of the changing back-ground. To apply the WKB approximation we define the param-eter δ as δ ≡ v L . (4)The validity of the WKB approximation is restricted to smallvalues of δ so as P δ ≪
1, where P is the period of the oscil-lations. In the observations by Okamoto et al. (2007), the meanflow velocity and period are v ≈
40 km s − and P ≈ L ∼ km these values result P δ ≈ . δ we define the dimensionless time, t ,as t = δ t , (5)and we express the solution to Equation (1) in the following form v r ( z , t ) = Q ( z , t ) exp (cid:18) i δ Ω ( t ) (cid:19) , (6)with Q ( z , t ) and Ω ( t ) functions to be determined. Next, wecombine Equations (1) and (6), and separate the di ff erent termsaccording to their order with respect to δ . As δ is small, the dom-inant terms are those with the lowest order in δ . We obtain twoequations for Q ( z , t ) and Ω ( t ) taking the terms with O (cid:16) δ (cid:17) and O (cid:16) δ (cid:17) , namely ∂ Q ( z , t ) ∂ z + ∂ Ω ( t ) ∂ t ! Q ( z , t ) v ( z , t ) = , (7) Q ( z , t ) ∂ Ω ( t ) ∂ t + ∂ Q ( z , t ) ∂ t ∂ Ω ( t ) ∂ t = . (8)Equations (7) and (8) are equivalent to Equations (24) and (25)of Morton & Erd´elyi (2010a), respectively.Now, we define the time-dependent frequency, ω ( t ), as ω ( t ) ≡ ∂ Ω ( t ) ∂ t , (9)which allows us to rewrite Equation (7) as follows ∂ Q ( z , t ) ∂ z + ω ( t ) v ( z , t ) Q ( z , t ) = . (10)Equation (10) has to be solved taking into account the boundaryconditions Q ( ± L / , t ) = z of function Q ( z , t )can be obtained. In addition, since v ( z , t ) is a piecewise con-stant function of z (see Equation (2)), the analytical solutions toEquation (10) are trigonometric functions with time-dependentarguments. Thus, the general solution to Equation (10) satisfyingthe boundary conditions at z = ± L / Q ( z , t ) = A ( t ) sin (cid:16) ω ( t ) v ke (cid:16) z + L (cid:17)(cid:17) if z < z − , A ( t ) cos (cid:18) ω ( t ) v kp z + φ ( t ) (cid:19) if z − ≤ z ≤ z + , A ( t ) sin (cid:16) ω ( t ) v ke (cid:16) z − L (cid:17)(cid:17) if z > z + (11) R. Soler and M. Goossens: Kink oscillations of flowing prominence threads
Fig. 2.
Dimensionless frequency, ωτ Ap , versus time in units ofthe internal Alfv´en travel time, τ Ap = L / v Ap . Results correspond-ing to the fundamental mode and the lowest seven harmonics ob-tained by numerically solving Equation (14) for a flowing threadwith L p / L = v / v Ap = z / L = − .
25. The verti-cal dotted line denotes the time when the prominence thread iscentered within the flux tube.where A ( t ), A ( t ), and A ( t ) are time-dependent coe ffi cients, φ ( t ) is a time-dependent phase, and z − and z + denote the loca-tions of the interfaces between the prominence thread and theevacuated regions, namely z ± = z ± L p + t L . (12)The locations of the interfaces change as the dense thread movesalong the magentic tube. Q must satisfy appropriate boundaryconditions at z = z ± . Since the interfaces correspond to con-tact discontinuities (see Goedbloed & Poedts 2004), the bound-ary conditions are[[ Q ]] = , "" ∂ Q ∂ z = , (13)where [[ X ]] stands for the jump of the quantity X at z = z ± .Applying the conditions of Equation (13) on the solutionsgiven by Equation (11), we arrive at the following equation v ke v kp tan " ω ( t ) v ke z − L − L p + t L ! = v kp + v ke cot (cid:18) ω ( t ) v kp L p (cid:19) tan h ω ( t ) v ke (cid:16) z + L − L p + t L (cid:17)i v kp cot (cid:18) ω ( t ) v kp L p (cid:19) − v ke tan h ω ( t ) v ke (cid:16) z + L − L p + t L (cid:17)i . (14)Equation (14) is the time-dependent dispersion relation. Forfixed t , the solution of Equation (14) is ω ( t ). Note that, al-though Equation (14) is written in a more compact form, it isconsistent with dispersion relations previously obtained for nor-mal modes in the static case, i.e., v =
0. Equation (14) with v = L + e → L − L p − z and L − e → L − L p + z are performedin their expression. Also for v =
0, Equation (14) is similar toEquation (17) of Joarder & Roberts (1992) and Equation (A5) ofOliver et al. (1993) obtained in Cartesian geometry.We have solved Equation (14) by standard numerical tech-niques. The frequencies of the fundamental mode and of the low-est seven harmonics with respect to z are displayed as functions of time in Figure 2 for a particular set of parameters. We findthat the dispersion diagram is symmetric with the time when thethread is located at the center of the magnetic tube (denoted bya vertical dotted line in Fig. 2) as point of symmetry. The fun-damental mode and the first harmonic are smooth functions oftime. The other harmonics displayed in Figure 2 show a com-plicated set of couplings and avoided crossings. The reason forthis behavior is that the fundamental mode and the first har-monic correspond to global oscillations of the flux tube becauseboth the prominence and the evacuated parts of the tube are dis-turbed. On the contrary, high harmonics correspond to modesmore confined within one of these regions. Thus, the collec-tion of modes and their properties are similar to those studiedby Joarder & Roberts (1992) and Oliver et al. (1993) in slab ge-ometry. From hereon we restrict our analysis to the fundamental modeof oscillation, whose frequency is the lowest order solution toEquation (14). To obtain an approximation to the frequency, weperform a Taylor expansion of Equation (14) and neglect termswith O (cid:16) ω (cid:17) and higher orders in ω . The following expression isobtained ω ( t ) ≈ v kp s LL p × r(cid:16) L − L p (cid:17) (cid:16) L + L p (cid:17) − z + t L ) (cid:18) + ρ e + ρ c ρ p + ρ c ( L − L p ) L p (cid:19) . (15)The e ff ect of the flow is contained in the denominator ofthe right-hand side of Equation (15). We see that the e ff ectof the flow on the frequency is more complicated than a sim-ple Doppler shift. There are two reasons that cause this depen-dence. On the one hand, our model is a complicated structurein the sense that only the dense prominence material is mov-ing. It is well known that a wave propagating in a uniform mag-netic tube with a constant siphon flow is a ff ected by a constantDoppler shift of the frequency due to the flow. However, thee ff ect of the flow is not so simple in more complicated con-figurations. Even in the case of a flux tube with a constantflow within the tube but no flow in the exterior of the tube thewave frequencies su ff er corrections due to the flow that are notsimple frequency shifts (see, e.g., Nakariakov & Roberts 1995;Terra-Homem et al. 2003). Our configuration is very di ff erentfrom the typical uniform magnetic flux tube with a siphon flow,so that the frequency is also modified by the change of positionof the dense plasma within the magnetic tube. On the other hand,we are dealing with standing modes, not propagating waves. Forstanding modes in flux tubes Terradas et al. (2011) have shownthat flow produces a spatially dependent phase shift along themagnetic tube. In our case, this phase shift is contained in thetime-dependent phase φ ( t ) of Equation (11).For typical prominence and coronal densities, ρ p ≫ ρ c and ρ p ≫ ρ e . Therefore, the term with the ratio of densitiesin the denominator of Equation (15) can be neglected. Then,Equation (15) simplifies to ω ( t ) ≈ v kp q LL p q(cid:16) L − L p (cid:17) (cid:16) L + L p (cid:17) − z + t L ) . (16) . Soler and M. Goossens: Kink oscillations of flowing prominence threads 5 In the absence of flow, i.e., t =
0, and for z =
0, Equation (16)loses its time dependence and becomes, ω ≈ v kp s L (cid:16) L − L p (cid:17) (cid:16) L + L p (cid:17) L p . (17)Equation (17) is consistent with the approximation of the normalmode frequency obtained by D´ıaz et al. (2010, Equation (8a)).For L p ≪ L we can approximate L + L p ≈ L , andEquation (17) reduces to the expression found by Soler et al.(2010, Equation (17)). Although the case L p → L is very un-realistic in prominences because the observed lengths of promi-nence threads correspond to L p / L ≪
1, it is instructive to takeinto account this limit. For L p → L the frequency given inEquation (17) tends to infinity. The reason is that the fundamen-tal kink mode behaves as a hybrid mode like those described byOliver et al. (1993) in a Cartesian slab (see also the string modes investigated by Joarder & Roberts 1992). Equations (15)–(17)are approximations of the hybrid mode frequency. As explainedby Oliver et al. (1993), hybrid modes owe their existence to thepresence of both the dense part and the evacuated part of thetube. In the limit L p → L the evacuated part is absent and the hybrid kink mode disappears. Thus for L p → L the fundamen-tal kink mode is not the hybrid mode but the first internal modewith frequency ω = v kp π L . (18)Since L p / L ≪ hybrid mode and Equations (15)–(17) apply.We plot in Figure 3 the solution of Equation (14) for the fun-damental mode as a function of time, along with the analyticalapproximation given by Equation (16). Equation (14) has beensolved by using standard numerical methods to obtain the rootsof transcendental equations. These computations have been per-formed for di ff erent values of L p / L and v / v Ap , and for fixed z / L . There is a very good agreement between the full solution(solid lines in Figure 3) and the approximation (symbols). FromFigure 3(a), we see that the frequency decreases as the length ofthe prominence thread increases. On the other hand, Figure 3(b)shows that the variation of the frequency with time is more im-portant as the flow velocity gets faster. We find that the mini-mum of the frequency takes place when the thread is centeredwithin the magnetic tube. Therefore, the minimum of the fre-quency depends on both the initial position of the thread and theflow velocity since the relation z + v t = Ω ( t ). We ob-tain Ω ( t ) from Equation (9) by integrating ω ( t ) given byEquation (16). Hence, Ω ( t ) = v kp p LL p arctan z + t L ) q(cid:16) L − L p (cid:17) (cid:16) L + L p (cid:17) − z + t L ) − arctan z q(cid:16) L − L p (cid:17) (cid:16) L + L p (cid:17) − z , (19)where we have used the condition Ω = t = Here, we estimate the variation of the amplitude of the os-cillations with time. To do so, we use Equation (8). By tak-ing into account the definition of the time-dependent frequency(Equation (9)), we rewrite Equation (8) as ∂ Q ( z , t ) ∂ t + ω ( t ) ∂ω ( t ) ∂ t Q ( z , t ) = . (20)Next, we consider the approximate ω ( t ) for the fundamentalmode obtained in Equation (15) to express this last Equation as ∂ Q ( z , t ) ∂ t + L ( z + t L ) (cid:16) L − L p (cid:17) (cid:16) L + L p (cid:17) − z + t L ) Q ( z , t ) = . (21)Note that to solve Equation (21) we do not have to care aboutthe dependence of Q on z . For a given z , Equation (21) can beintegrated to obtain the temporal dependence of Q at a fixedposition, namely Q ( z , t ) = Q ( z ) (cid:16) L − L p (cid:17) (cid:16) L + L p (cid:17) − z + t L ) (cid:16) L − L p (cid:17) (cid:16) L + L p (cid:17) − z / , (22)where Q ( z ) is the amplitude at t =
0. Thus, Equation (11)gives the spatial dependence of Q for a fixed time t , whileEquation (22) provides the temporal dependence at a fixed po-sition z . By comparing Equations (16) and (22), we see that Q ( z , t ) ∝ ω ( t ) − / . Then, the amplitude of the oscillation de-creases when the thread flows from the center of the tube to thefootpoint, and increases otherwise. This means that the ampli-tude is maximal when the thread is located at the center of themagnetic tube.We must bear in mind that Equation (22) was derived fromthe terms with O (cid:16) δ (cid:17) in the governing equation, whereas theleading terms when δ is a small parameter are those with O (cid:16) δ (cid:17) .Therefore, we have to be cautious about the actual accuracy ofEquation (22), although we expect the behavior of the amplitudewith time to be, at least, qualitatively described by Equation (22). The analytical expression of the fundamental mode frequencyfound before can be used to perform magneto-seismology ofprominence fine structures by using observed periods of oscil-lations in flowing threads. We use Equation (16) to compute theperiod of the oscillation as a function of time, namely P ( t ) = πω ( t ) ≈ π v kp r L p L s(cid:16) L − L p (cid:17) L + L p ! − z + v t ) , (23)which we have explicitly written in terms of the dimensionaltime, t , and the flow velocity, v . By using Equation (23) alongwith observational values of the period, it is possible to give anestimation of L , i.e., the total length of the flux tube, which isa parameter di ffi cult to measure from the observations. Let usassume that we have performed an observation of a transverselyoscillating and flowing thread with a good cadence and we havedetermined the evolution of the period with time. For conve-nience, we set t = R. Soler and M. Goossens: Kink oscillations of flowing prominence threads
Fig. 3.
Dimensionless frequency, ωτ Ap , versus time in units of the internal Alfv´en travel time, τ Ap = L / v Ap , for (a) L p / L = v / v Ap = v / v Ap = L p / L = .
1. In all cases, z / L = − .
25. The solid lines are theresults obtained by numerically solving Equation (14), whereas the symbols correspond to the analytical approximation given byEquation (16). The vertical dotted line in panel (a) denotes the minimum of the curves, i.e., the time when the prominence thread iscentered within the flux tube.place, namely P (0), so we can also fix z = P ( τ ) the instantaneous period measured at t = τ . We use Equation (23) and compute the ratio P ( τ ) / P (0)to find an estimation of the length of the magnetic flux tube as L ≈ L p + v τ L p − (cid:16) P ( τ ) P (0) (cid:17) , (24)where again we assumed that the flow velocity is slow. We seethat the right-hand side of Equation (24) depends on quantitiesthat can be directly measured from the observations. As an ex-ample, let us assume the following values: P ( τ ) / P (0) = . v =
40 km s − , L p = ,
000 km, and τ = L ≈ . × km. However, the ac-curacy of Equation (24) is limited by the uncertainties and errorbars of the observations. In particular, a very accurate determina-tion of the ratio P ( τ ) / P (0) is needed. For instance, a 10% un-certainty of P ( τ ) / P (0) produces a 85% uncertainty of L whenpropagation of errors is used in Equation (24) and the remainingparameters are kept constant. This makes a reliable applicationof Equation (24) di ffi cult in practice.Another relevant parameter that can give us a seismologicaldetermination of L is the ratio P / P , where P is the periodof the first harmonic. The deviation of this ratio from unity is anindication of the longitudinal inhomogeneity length scale of themagnetic tube. Its application was used for the first time in thecontext of coronal loop oscillations by Andries et al. (2005a,b)and has been explored in subsequent works (see the recent re-view by Andries et al. 2009, and references therein). In promi-nence thread oscillations, D´ıaz et al. (2010) explored the impor-tance of the ratio P / P to estimate L p / L in static threads.While in coronal loops P / P <
1, D´ıaz et al. (2010) foundthat in prominence threads P / P >
1. The reason for this re-sult is that in prominence threads mass density is arranged justin the opposite way to that in coronal loops. In loops the densityis larger in the footpoints than in the apex due to gravitationalstratification, while in prominence threads the density is muchlarger in the center of the magnetic tube because of the presenceof the prominence material.
Fig. 4.
Ratio P / P of a flowing thread with L p / L = . z / L = − .
25 for v / v Ap = P / P and L p / L obtained by D´ıaz et al. (2010) in their Equation (11) is P P ≈ s L p / L . (25)Let us see how the ratio P / P is a ff ected by the flow andso how the results of D´ıaz et al. (2010) are modified. However,it is di ffi cult to obtain an analytical expression for P whenflow is present. Instead, we compute both P and P by solvingEquation (14) with numerical methods. Figure 4 shows P / P as a function of time for L p / L = . z / L = − .
25, and fordi ff erent values of v / v Ap . The numerical results are comparedwith the analytical expression of D´ıaz et al. (2010). First of all,we see that the period ratio is strongly influenced by the veloc- . Soler and M. Goossens: Kink oscillations of flowing prominence threads 7 ity at which the prominence thread flows along the flux tube. P / P is maximal when the thread in centered within the tube( P / P ≈ . P / P → P / P in comparison to the case withflow. This means that flow reduces the period ratio. Therefore,Equations (24) and (25) could be used together to obtain a moreaccurate determination of the magnetic tube length, as the valueof L inferred from Equation (25) should be considered as a upperbound for this parameter.
4. Numerical results: time-dependent simulations
Here, we compare the analytical results of the WKB approxi-mation with the full numerical solution of the time-dependentproblem. We use the PDE2D code (Sewell 2005) for that pur-pose. The set-up of the numerical code is similar to that ofTerradas et al. (2008). Equation (1) is integrated assuming theboundary conditions v r ( ± L / , t ) =
0. An initial condition for v r at t = L and the velocities in units of the prominenceAlfv´en speed, v Ap = B √ µρ p . We take L = km. Assuming B =
50 G and ρ p = − kg m − as realistic values in activeregion prominences, we obtain v Ap ≈
446 km s − . The flow ve-locities on the plane of sky estimated by Okamoto et al. (2007)are in the interval between 15 km − to 46 km s − . Hence in oursimulations we consider values for the ratio v / v Ap in the range0 . . v / v Ap . .
1. In the code, time is expressed in units ofthe Alfv´en travel time, i.e., τ Ap = L / v Ap ≈ .
74 min. In addition,in all the following computations we have used ρ p /ρ c =
200 and ρ e /ρ c = First, we use the eigenfunction of the fundamental kink mode asthe initial condition for v r at t =
0. The eigenfunction is obtainedby solving the dispersion relation of the normal mode problemand computing the spatial distribution of the corresponding per-turbation (see details in Dymova & Ruderman 2005; Soler et al.2010; D´ıaz et al. 2010). Hence, we make sure that, after the ini-tial excitation, the magnetic tube mainly oscillates in its funda-mental mode.As a check of the numerical code, we consider the static caseand put the thread at the center of the magnetic tube, i.e., v = z =
0. In this test simulation, we take L p / L = .
1. Bylooking at the time-dependent evolution of v r , we check that themagnetic tube oscillates as a whole. A plot of v r at z = v r at z = P ≈ . ff erent values of L p / L and z . Equivalent results to those commented before havebeen obtained in all cases. This indicates, on the one hand, thatthe normal mode interpretation is a very good representation for the time-dependent evolution of the oscillation in the static caseand, on the other hand, that the numerical code works properly.Hereafter, we incorporate the e ff ect of the flow. First, we fix L p / L = . z =
0, and consider v / v Ap = .
05. In this situ-ation, the thread is initially located at the center of the magnetictube. We make sure that the simulation stops before the threadsreaches the photospheric wall. We plot in Figure 5(a) the radialvelocity perturbation at z = Q ( z , t ) = Q ( z ) (cid:16) L − L p (cid:17) (cid:16) L + L p (cid:17) − z + t L ) (cid:16) L − L p (cid:17) (cid:16) L + L p (cid:17) − z n , (26)with n an empirical exponent. When Equation (26) is appliedto the results of Figure 5(a), we obtain that the exponent n = t = z = L / z =
0, whereas the period decreases of about 45% whenthe thread finally reaches the footpoint of the magnetic tube at z = L .We repeat the simulation for L p / L = . v / v Ap = .
1, and take z / L = − .
25 to consider the threadinitially displaced from the center of the flux tube. The radialvelocity perturbation at z = In the previous Section, we have used an initial condition for v r that corresponds to the fundamental mode eigenfunction, soonly this mode is excited. However, it is expected that the en-ergy from an arbitrary disturbance of the flux tube is depositedin many normal modes (see a discussion on this issue in, e.g.,Terradas et al. 2007). To represent an arbitrary disturbance of the R. Soler and M. Goossens: Kink oscillations of flowing prominence threads
Fig. 5. (a) v r at z = L p / L = . v / v Ap = . z =
0. The initial excitation is the fundamental kink mode eigenfunction for v =
0. The dashed line is the amplitude in theWKB approximation (Equation (22)), while the dotted line corresponds to the fit proposed in Equation (26) with n =
1. (b) Waveletpower spectrum for the dimensionless period, P /τ Ap , corresponding to the signal displayed in panel (a). The white dashed line isthe period in the WKB approximation (Equation (23)), whereas the horizontal dotted line is the period at t =
0. The red solid linedenotes 99% of confidence level. (c) Same as panel (a) but for L p / L = . v / v Ap = .
1, and z / L = − .
25. (d) Same as panel (b)but for the signal of panel (c), and with the horizontal dotted line denoting the maximum of the period.flux tube, we consider a Gaussian function as the initial condi-tion of v r at t =
0, namely v r ( z , t = = exp " − ( z − ζ ) σ , (27)where ζ and σ are arbitrary parameters. Whereas ζ correspondto the position of the maximum of the excitation, σ determinesits width.In the following simulations, we consider the same modelparameters as in Figures 5(c)–(d), i.e., L p / L = . v / v Ap = .
1, and z / L = − .
25, but use the initial condition given byEquation (27). To begin with, we take σ/ L = . ff erent values of ζ . First we use ζ/ L = − .
25, so the excitationis mainly confined to the dense prominence region of the fluxtube. The result of this simulation is displayed in Figure 6(a),which shows the evolution in time of v r at z =
0, whereasFigure 6(b) shows the corresponding wavelet power spectrum.It is interesting to compare Figures 5(d) and 6(b) to see that, inthe present case, the oscillation dynamics is still governed bythe fundamental normal mode. We see in Figure 6(a) that thereis some contribution of higher harmonics to the behavior of v r intime, although their contribution to the overall oscillation is ofvery minor importance. In addition, the evolution of the ampli-tude of v r remains qualitatively described by Equation (26) with n = ζ/ L =
0. In this case, the max-imum of the initial excitation is located in the evacuated part of the magnetic tube. Again, we plot in Figure 6(c) v r at z = v r in time is substantially di ff erent in the presentsituation compared to the case of Figures 6(a)–(b). First of all,we see that v r is not governed by the fundamental mode ex-clusively. The wavelet power spectrum indicates that the energyfrom the initial disturbance is mainly deposited to the fundamen-tal mode, but also the first harmonic is excited. The dependencein time of the fundamental mode period is again well describedby Equation (23). However, the contribution of the first harmonicto the overall oscillation seems to have disappeared when thethread is located at the center of the magnetic tube. The reasonfor this result is that the first harmonic eigenfunction has a nodeat z = ffi cult to determine thee ff ect of the flow on the amplitude of the oscillation.Finally, we have performed several simulations for di ff erentvalues of σ . If the maximum of the excitation is located withinthe dense part of the flow tube, the results are rather insensitive to σ unless values much smaller than L p are used. In all the cases,the fundamental mode is predominantly excited. However, theresults are more a ff ected by the value of σ if the maximum ofthe excitation is located in the evacuated part of the tube. In sucha case, the larger σ , the more energy is deposited in the funda-mental mode. On the contrary, as σ gets smaller the energy of theinitial excitation is more distributed among higher harmonics. . Soler and M. Goossens: Kink oscillations of flowing prominence threads 9 Fig. 6. (a) Same as Figure 5(c) but for a initial excitation given by Equation (27) with ζ/ L = − .
25 and σ/ L = .
2. (b) Waveletpower spectrum corresponding to the signal displayed in panel (a). (c) Same as panel (a) but for ζ/ L = σ/ L = .
2. (d) Sameas panel (b) but for the signal displayed in panel (c), with the dash-dotted line denoting the period of the first harmonic.The results of this Section point out that the time-dependentbehavior of standing kink MHD waves of flowing prominencethreads is strongly influenced by the form of the initial distur-bance. If the initial disturbance mainly perturbs the dense promi-nence part of the flux tube, the oscillations are governed by thefundamental kink mode. In such a case, the dependence of boththe period and the amplitude with the flow velocity are approx-imately given by Equations (23) and (26), respectively. On thecontrary, the behavior is more complicated if the initial pertur-bation takes place in the evacuated part of the fine structure asother harmonics are excited in addition to the fundamental mode.The contribution of the di ff erent harmonics depends on both theposition and the width of the initial excitation, while the ampli-tude of the oscillation does not have a simple dependence on theflow velocity.
5. Discussion and conclusions
In this paper, we have investigated standing kink MHD waves inthe fine structure of solar prominences, modeled as coronal mag-netic flux tubes partially filled with flowing threads of promi-nence material. The present study extends and complements theprevious work by Terradas et al. (2008), who restricted them-selves to the numerical investigation of this phenomenon anddid not perform an in-depth parametric study. Here, we havecombined analytical methods based on the WKB approximationwith time-dependent numerical simulations to assess the precisee ff ect of the flow on both the period and the amplitude of thefundamental kink mode.As for the e ff ect of the flow on the period, we can distinguishtwo di ff erent situations. On the one hand, we find that the flowhas a small e ff ect on the period when the thread is located near the center of the supporting magnetic flux tube. In this case, thevariation of the period with respect to the static case may fallwithin the error bars of the observations, and so the e ff ect maybe undetectable. There our results confirm the qualitative discus-sion of Terradas et al. (2008) about the e ff ect of the flow on theperiod. On the other hand, the variation of the period is muchmore important when the thread approaches the footpoint of themagnetic structure. Then, the decrease of the period can be largerthan 50% with respect to the static case. The case in which thethread is near one of the footpoints of the magnetic tube was notanalyzed by Terradas et al. (2008).We have also found that the flow a ff ects the amplitudeof the fundamental mode. This result was not discussed byTerradas et al. (2008). During the motion of the prominencethread along the magnetic structure, we find that the ampli-tude grows as the thread gets closer to the center of the tubeand decreases otherwise. This produces an apparent amplifica-tion or damping of the oscillations, respectively. Observationsoften indicate that thread transverse oscillations are stronglydamped (see, e.g., Lin 2004, 2010; Ning et al. 2009). While sev-eral mechanisms have been proposed and investigated to ex-plain the quick attenuation (see the recent reviews by Oliver2009; Arregui & Ballester 2010), the process of resonant absorp-tion seems the most likely explanation (e.g., Arregui et al. 2008;Soler et al. 2009a,b, 2010). Our present results indicate that theactual damping rate of the oscillations might be a ff ected by thechange of the amplitude due to the flow. This fact should betaken into account when the damping rate is used as a seismo-logical tool to infer physical parameters of prominence threads,because the presence of flow may introduce some uncertaintieson these estimations (see details in Arregui & Ballester 2010). In addition, our numerical simulations have allowed us to de-termine how di ff erent perturbations excite the oscillations of themagnetic structure. Based on the cases studied in this paper, wehave obtained that the fundamental mode is mostly excited whenthe perturbation initially disturbs the dense, prominence part ofthe tube. From the wavelet power spectrum of the radial velocityperturbation, we conclude that the contribution of higher har-monics is negligible, thus the overall oscillation is governed bythe fundamental mode. On the contrary, a perturbation located atthe evacuated part of the tube excites the fundamental mode andhigher harmonics, producing a more complex behavior of the os-cillations. In this last case, the e ff ect of the flow on the amplitudeis more complicated and no simple dependence can be extractedfrom the simulations.This paper has explored the properties of MHD waves ina coronal magnetic structure with a changing configuration.Previous similar works in this line are, e.g., Terradas et al. (2008)in prominences, and Morton et al. (2010); Morton & Erd´elyi(2010a,b) in coronal loops. During the revision of this paperit also came to our knowledge the recent work by Ruderman(2011). In view of the highly dynamic nature of the coronalmedium in general, and the prominence plasma in particular, thiskind of modeling represents a better description of the actualoscillatory phenomena in the corona and in prominences. Thepresent investigation could be extended in the future by incorpo-rating the e ff ect of the density inhomogeneity in the transversedirection and so investigating the resonant damping of the kinkmode. Acknowledgements.
RS thanks J. L. Ballester, R. Oliver, and T. VanDoorsselaere for useful comments. RS acknowledges support from a Marie CurieIntra-European Fellowship within the European Commission 7th FrameworkProgram (PIEF-GA-2010-274716). RS also thanks support from the EUResearch and Training Network “SOLAIRE” (MTRN-CT-2006-035484). RSacknowledges discussion within ISSI Team on “Solar Prominence Formationand Equilibrium: New data, new models”, and is grateful to ISSI for the finan-cial support. MG acknowledges support from K.U. Leuven via GOA / // paos.colorado.edu / research / wavelets / References
Ahn, K., Chae, J., Cao, W., & Goode, P. R. 2010, ApJ, 721, 74Andries, J., Goossens, M., Hollweg, J. V., Arregui, I., & Van Doorsselaere, T.2005a, A&A, 430, 1109Andries, J., Arregui, I., & Goossens, M. 2005b, ApJ, 624, L57Andries, J., Van Doorsselaere, T., Roberts, B., Verth, G., Verwichte, E., &Erd´elyi, R. 2009, Space Sci. Rev., 149, 3Antolin, P., & Verwichte, E. 2011, ApJ, in press (arXiv:1105.2175)Arregui, I., Terradas, J., Oliver, R., & Ballester, J. L. 2008, ApJ, 682, L141Arregui, I., & Ballester, J. L. 2010, Space Sci. Rev., in press(DOI:10.1007 / s11214-010-9648-9)Ballester, J. L. 2006, Phil. Trans. R. Soc. A, 364, 405Bender, C. M., & Orszag, S. A. 1978, Advanced Mathematical Methods forScientists and Engineers (New York: McGraw-Hill)Berger et al. 2008, ApJ, 676, L89Cao, W., Ning, Z., Goode, P. R., Yurchyshyn, V., & Haisheng, J. 2010, ApJ, 719,L95Carbonell, M., & Ballester, J. L. 1991, A&A, 249, 295Chae, J., Ahn, K., Lim, E.-K., Choe, G. S., & Sakurai, T. 2008, ApJ, 689, L73Chae, J. 2010, ApJ, 714, 618De Pontieu, B., & McIntosh, S. W. 2010, ApJ, 722, 1013D´ıaz, A. J., Oliver R., Erd´elyi, R., & Ballester, J. L.2001, A&A, 379, 1083D´ıaz, A. J., Oliver R., & Ballester, J. L. 2002, ApJ, 580, 550D´ıaz, A. J., Oliver R., & Ballester, J. L. 2010, ApJ, 725, 1742Dymova, M. V., & Ruderman, M. S. 2005, Sol. Phys., 229, 79Edwin, P. M., & Roberts, B. 1983, Sol. Phys., 88, 179Erd´elyi, R., & Fedun, V. 2007, Science, 318, 1572Goedbloed, H., & Poedts, S. 2004, Principles of magnetohydrodynamics,Cambridge University Press Goossens, M., Terradas, J., Andries, J., Arregui, I., Ballester, J. L. 2009, A&A,503, 213Jess, D. B., et al. 2009, Science, 323, 1582Joarder, P. S. & Roberts, B. 1992, A&A, 261, 625Joarder, P. S., Nakariakov, V. M., & Roberts, B. 1997, Sol. Phys., 173, 81Kucera, T. A., Tovar, M., & De Pontieu, B. 2003, Sol. Phys., 212, 81Lin, Y., Engvold, O., & Wiik, J. E. 2003, Sol. Phys., 216, 109Lin, Y. 2004, PhD Thesis, University of Oslo, NorwayLin, Y., Martin, S. F., & Engvold, O. 2008, in ASP Conf. Ser. 383, Subsurfaceand Atmospheric Influences on Solar Activity, ed. R. Howe, R. W. Komm, K.S. Balasubramaniam, & G. J. D. Petrie (San Francisco: ASP), 235Lin, Y., Soler, R., Engvold, O., Ballester, J. L., Langangen, Ø., Oliver, R., &Rouppe van der Voort, L. H. M. 2009, ApJ, 704, 870Lin, Y. 2010, Space Sci. Rev., in press (DOI:10.1007 / s11214-010-9672-9)Morton, R. J., Hood, A. W., & Erd´elyi, R. 2010, A&A, 512, A23Morton, R. J., & Erd´elyi, R. 2010a, ApJ, 707, 750Morton, R. J., & Erd´elyi, R. 2010b, A&A, 519, A43Nakariakov, V. M., & Roberts, B. 1995, Sol. Phys., 159, 213Ning, Z., Cao, W., Okamoto, T. J., Ichimoto, K., & Qu, Z. Q. 2009, A&A, 499,595Ofman, L., & Wang, T. J. 2008, A&A, 482, L9Okamoto, T. J, et al. 2007, Science, 318, 1557Oliver, R., Ballester, J. L., Hood, A. W., & Priest, E. R. 1993, ApJ, 409, 809Oliver, R. 2009, Space Sci. Rev., 149, 175Ruderman, M. S. 2011, Sol. Phys., in press (DOI:10.1007 / s11207-011-9772-z)Schmieder, B., Chandra, R., Berlicki, A., & Mein, P. 2010, A&A, 514, A68Sewell, G. 2005, The Numerical Solution of Ordinary and Partial Di ffff