On the linear coupling between fast and slow MHD waves due to line-tying effects
aa r X i v : . [ a s t r o - ph . S R ] N ov Astronomy&Astrophysicsmanuscript no. draftA˙A c (cid:13)
ESO 2018November 2, 2018
On the linear coupling between fast and slow MHD wavesdue to line-tying effects
J., Terradas , J., Andries , , E., Verwichte Departament de F´ısica, Universitat de les Illes Balears, E-07122, Spain, e-mail: [email protected] Centre Plasma Astrophysics and Leuven Mathematical Modeling and Computational Science Centre, Katholieke UniversiteitLeuven, Leuven, B-3001, Belgium, e-mail: [email protected] Centre for Stellar and Planetary Astrophysics, Monash University, Victoria, 3800, Australia Centre for Fusion, Space and Astrophysics, Department of Physics, University of Warwick, Coventry CV4 7AL, UK, e-mail:
Received / Accepted
ABSTRACT
Context.
Oscillations in coronal loops are usually interpreted in terms of uncoupled magnetohydrodynamic (MHD) waves. Examplesof these waves are standing transverse motions, interpreted as the kink MHD modes, and propagating slow modes, commonly reportedat the loop footpoints.
Aims.
Here we study a simple system in which fast and slow MHD waves are coupled. The goal is to understand the fingerprints ofthe coupling when boundary conditions are imposed in the model.
Methods.
The reflection problem of a fast and slow MHD wave interacting with a rigid boundary, representing the line-tying e ff ect ofthe photosphere, is analytically investigated. Both propagating and standing waves are analysed and the time-dependent problem ofthe excitation of these waves is considered. Results.
An obliquely incident fast MHD wave on the photosphere inevitably generates a slow mode. The frequency of the generatedslow mode at the photosphere is exactly the same as the frequency of the incident fast MHD mode, but its wavelength is much smaller,assuming that the sound speed is smaller than the Alfv´en speed.
Conclusions.
The main signatures of the generated slow wave are density fluctuations at the loop footpoints. We have derived asimple formula that relates the velocity amplitude of the transverse standing mode with the density enhancements at the footpointsdue to the driven slow modes. Using these results it is shown that there are possible evidences in the observations of the couplingbetween these two modes.
Key words.
Magnetohydrodynamics (MHD) — Waves — Magnetic fields — Sun: atmosphere—Sun: oscillations
1. Introduction
In coronal loops there is evidence of the presence of fast stand-ing magnetohydrodynamic (MHD) modes and slow (propa-gating and standing) MHD modes. Standing kink oscillationswere first reported using TRACE by Aschwanden et al. (1999)and Nakariakov et al. (1999). Later, similar observations wereanalysed by, e.g., Schrijver & Brown (2000); Aschwanden et al.(2002); Schrijver et al. (2002). There is also clear evidence ofthe presence of propagating slow waves at loop footpoints (seeDe Moortel, 2009, for a review). These slow waves are mostlikely due to coupling between the underlying atmospheric lay-ers since the dominant periods tend to be around 5 min, sug-gesting a possible link with solar p -modes. Furthermore, obser-vations of standing slow modes in coronal loops, with periodslargely above 5 min, have been also reported by a number of au-thors, using di ff erent instruments such as SOHO / SUMER (e.g.,Wang et al., 2003a,b, 2007), Yohkoh / BCS (e.g., Mariska, 2005,2006) and more recently, Hinode / EIS (e.g., Erd´elyi & Taroyan,2008).The theoretical interpretation of such a variety of oscilla-tions is done, in most of the cases, in terms of uncoupled MHDwaves. On one hand, transverse oscillations are identified as fast
Send o ff print requests to : J. Terradas, e-mail: [email protected] kink modes in the zero- β approximation, meaning that in the lin-ear regime there is no longitudinal velocity along the magnetictube. On the other hand, the reported slow modes are associ-ated to acoustic modes, ignoring the coupling with the transversemotions. Although these identifications are useful and simple,we have to bear in mind that under realistic conditions MHDmodes may couple. In general, the e ff ect of gas pressure, in-homogeneities or boundary conditions lead to the coupling be-tween fast and slow modes. As we shall see, a clear example ofcoupling comes from the line-tying condition, which amounts toconsidering the photosphere as providing a complete reflectionof any coronal disturbance impinging from above. This is justi-fied in most of the cases due to the large di ff erence in densitiesbetween the photosphere and corona but it obviously neglectsthe important role of the transition region.In this paper we study the simplest configuration in whichfast and slow modes are mixed to understand the basics ofmode coupling. We use the simple idea that a fast wave that isobliquely incident on a boundary will generate a slow wave. Asfar as we know, this issue has been addressed in slightly di ff er-ent contexts by, e.g., Stein (1971); Vasquez (1990); Oliver et al.(1992). A remarkable work about magnetohydrodynamic wavesin coronal flux tubes including the line-tying e ff ect was carriedout by Goedbloed & Halberstadt (1994). In that work, it wasclearly stated that pure fast or pure slow modes do not exist in Terradas et al.: Coupling between fast and slow MHD waves a line-tied coronal loop. Here we follow a di ff erent approach toanalyse this problem. Our interest is in the e ff ects of the applica-tion of line-tying conditions on fast and slow modes and in thepossible observational fingerprints of this coupling.
2. Model and Dispersion relation
We consider first the simplest magnetic configuration where themagnetic field points in the z -direction, the plasma density andpressure are constant, and v A > c s ( v A is the Alfv´en speed and c s the sound speed). Since the medium is homogeneous, we con-sider perturbations that are proportional to e i ( ω t + k x x + k z z ) , meaningthat waves propagate in the negative x and z -directions. In thisconfiguration, the linearised MHD equations (see Appendix A)lead to the well known dispersion relation for fast and slowMHD waves, ω − (cid:16) k x + k z (cid:17) (cid:16) c + v (cid:17) ω + k z (cid:16) k x + k z (cid:17) c v = . (1)In our problem it is more convenient to assume that ω and k x are known and solve for k z . The reason is that the frequency ofan incoming wave remains constant in the reflection problemwhile the longitudinal wavenumber can change. The dispersionrelation written as a biquadratic equation for k z is, c v k z − h(cid:16) c + v (cid:17) ω − c v k x i k z + ω h ω − k x (cid:16) c + v (cid:17)i = . (2)We denote by k F and k S the fast and the slow longitudinalwavenumbers that are solutions to the previous biquadratic equa-tion, i.e., k = − B − √ B − AC A , (3) k = − B + √ B − AC A , (4)where A = c v , (5) B = − h(cid:16) c + v (cid:17) ω − c v k x i , (6) C = ω h ω − k x (cid:16) c + v (cid:17)i . (7)These wavenumbers are associated to the same frequency ω andsame k x , and can be approximated, using the small plasma- β as-sumption, by (e.g. Oliver et al., 1992) k ≃ ω v − k x c v + , (8) k ≃ ω c . (9)These approximations will turn out to be useful in the followingsections. According to Eq. (8) the fast wavenumber may becomepurely imaginary for a certain choice of the parameters, indicat-ing that the wave is evanescent. Hereafter, we will restrict ouranalysis to propagating waves.It can be seen from the linearised MHD equations that thevelocity polarisation for fast and slow modes is given by the fol-lowing expression v z = c k x k z ω − k z c v x , (10) where k z is either k F or k S . Note that if we change the directionof propagation along the field, k z changes to − k z and the polari-sation also changes sign. In our configuration and in the regimethat we are interested ( v A > c s ), since k F < k S , fast modes arecharacterised by long wavelengths and by v x > v z , while slowmodes have short wavelengths and v x < v z . The slow modeshave a larger compression since the wavelength is small in com-parison with the fast modes.
3. Propagating waves: the reflection problem
We consider the most elementary problem to show the processof mode coupling due to boundary conditions. A fast MHD wavetravels downwards (in the negative z -direction) and represents anincoming wave. This wave interacts with the photosphere (lo-cated at z =
0, where line-tying conditions are applied) and re-flects (now travelling upwards). The amplitude of the fast incom-ing wave is F I , while the amplitude of the reflected fast wave is F R . To satisfy the boundary conditions a slow MHD wave, alsomoving upwards, must be generated at z =
0. The excited slowmode has an amplitude S G . It is easy to write the velocity com-ponents using the polarisation of fast and slow waves, given byEq. (10), and the proper sign of the longitudinal wavenumber, V x = F I e i ( ω t + k x x + k F z ) + F R e i ( ω t + k x x − k F z ) − S G ω − k c c k x k S e i ( ω t + k x x − k S z ) , (11) V z = F I c k x k F ω − k c e i ( ω t + k x x + k F z ) − F R c k x k F ω − k c e i ( ω t + k x x − k F z ) + S G e i ( ω t + k x x − k S z ) . (12)Now boundary conditions are imposed at z =
0, representing thelocation of the photosphere. We use line-tying conditions, i.e., V x ( z = = V z ( z = =
0. According to Eqs. (11) and (12) thefollowing conditions must be satisfied, F I + F R − S G ω − k c c k x k S = , (13) F I − F R + S G ω − k c c k x k F = . The solution in terms of the amplitude of the incoming wave(which is an arbitrary parameter) is F R F I = (cid:16) ω − k c (cid:17) k F + (cid:16) ω − k c (cid:17) k S (cid:16) ω − k c (cid:17) k F − (cid:16) ω − k c (cid:17) k S , (14) S G F I = c k x k S k F (cid:16) ω − k c (cid:17) k F − (cid:16) ω − k c (cid:17) k S . (15)The coe ffi cients depend on ω , k x (same for the three waves) andon longitudinal wavenumbers, k F and k S , which can also be ex-pressed in terms of ω , k x by using Eqs. (3) and (4).When there is no coupling between fast and slow modes (i.e.,when k x =
0) we have that | F R / F I | = ω = k c ,while S G / F I =
0. Hence, there is a complete reflection of theincoming fast mode while the slow mode is absent. When thereis coupling, the reflection coe ffi cient for the slow mode is alwaysdi ff erent from zero, meaning that the incoming fast MHD wave erradas et al.: Coupling between fast and slow MHD waves 3 Fig. 1.
Amplitude of reflected fast mode given by Eq. (14) as a functionof the horizontal wavenumber for three di ff erent values of the soundspeed. inevitably generates a slow mode at the boundary. In Figure 1the reflection coe ffi cient of the fast wave, i.e., F R / F I , is plot-ted as a function of k x for di ff erent values of the sound speed.The coe ffi cient is close to unity for values of sound speed lowin comparison with the Alfv´en speed, meaning that reflection isalmost total. However, for relatively large values of c s / v A thecurves clearly show a minimum indicating that the reflection ofthe fast wave is less e ffi cient. Fig. 2.
Amplitude of the generated slow mode given by Eq. (15) asa function of the horizontal wavenumber for three di ff erent values ofthe sound speed. The dashed line represents the approximation givenby Eq. (18) while the dotted line shows the position of the maximaaccording to Eqs. (19) and (20). In Figure 2 the reflection coe ffi cient of the slow wave, i.e., S G / F I , is plotted. The behaviour of the reflected slow modeamplitude is basically opposite of the reflected fast wave. Thecurves show a maximum at the locations where the generationof the slow mode is more e ffi cient. Again the value at the max-imum strongly depends on the ratio of the sound speed to theAlfv´en speed, and corresponds to a k x which is very similar to k F . The location of the maximum can be analytically determinedfrom Eq. (15) and using the approximation of the slow mode frequency, ω ≃ k c , S G F I ≃ k x k F k − k . (16)Since the frequency is fixed, we have from the approximate ex-pressions (8)-(9) for k and k the following relation k ≃ k x + v c + k v c . (17)Inserting this expression in Eq. (16) we find that S G F I ≃ c k x k F k (cid:16) c − v (cid:17) − k x (cid:16) c + v (cid:17) . (18)This is a good approximation to the slow mode reflection coe ffi -cient (see dashed line in Figure 2), at least for the case c s ≪ v A .It is used to determine the location of the maxima of the curvesin Figure 2. Imposing that the derivative of Eq. (18) is equal tozero it is found that the maximum corresponds to k x max ≃ k F s v − c v + c , (19)and the value of the reflection coe ffi cient for this horizontalwavenumber is (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) S G F I (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) max ≃ c q v − c . (20)In Figure 2 the location and value of the maximum is representedby a dotted line. There is excellent agreement between the ap-proximation and the location of the maximum calculated usingthe original expression for the reflection coe ffi cient of the slowmode (Eq. (15)). From Eq. (20) we see that the maximum am-plitude of the generated slow wave depends only on the β of theplasma.Once we know the coe ffi cients it is straight forward to calcu-late magnitudes of interest such as the density perturbation of theslow reflected wave as a function of amplitude of the incomingfast wave. As we will discuss later, these magnitudes can be di-rectly related to real observations. From the continuity equation(see Appendix A) the density perturbation for the slow mode is (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ρ ρ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = ω ( k x V x + k z V z ) = k S ω − ω − k c k c S G . (21)Using again the fact that ω ≃ k c and Eq. (20) we find (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ρ ρ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) max ≃ c s F I q v − c . (22)This equation relates the maximum density perturbation associ-ated to the generated slow mode with the velocity amplitude ofthe incident fast wave. The time-dependent problem is numerically solved to demon-strate the mode coupling described in the previous section. Thetemporal evolution of the waves and specially their interactionwith the boundaries provides a clear picture of the coupling andcomplements the analytical results.
Terradas et al.: Coupling between fast and slow MHD waves
In this numerical experiment a pulse in the v x component isgenerated at t =
0. This pulse has the following form v x ( z , t = = v cos ( kz ) e − (cid:16) z − z a (cid:17) , (23) v z ( z , t = = , (24)and the rest of the perturbed variables are set to zero. Thisparticular profile has the property that represents a localisedwave packet but with a rather well defined wavelength (basicallygiven by λ = π/ k ). To show the interaction of the fast modewith a single boundary, rigid conditions are applied at z = v x ( z = = v z ( z = =
0) while open conditions are imposedat z = L . The linearised time-dependent MHD equations (seeAppendix A) are numerically solved using standard finite di ff er-ences techniques.The temporal evolution of the two velocity components isplotted in Figure 3. The initial disturbance excites the fast modebut also the slow mode since the initial perturbation does notsatisfy the velocity polarisation relation, given by Eq. (10). Theexcited fast and slow modes split in two identical modes prop-agating in opposite directions (top panel). The fast and slowwaves travelling to the left are denoted as F I and S I , respec-tively, while F I + and S I + move to the right. Once the F I modereaches the rigid boundary ( z = F R and a slow mode is generated(see dotted line), i.e., the S G mode according to the notation in-troduced above. The slow wave packet has a typical wavelengthmuch smaller than the fast wavelength ( k F ≪ k S ). At the otherboundary ( z = L ) the fast mode F I + leaves the system due to thetransparent boundary conditions. At later times (bottom panel)the S G and S I modes, moving in opposite directions, start to su-perpose, while the F R and S I + , travelling in the same directionalso interfere.The amplitudes of reflected fast and slow waves estimatedfrom the time-dependent problem agree quite well with the cal-culations based on the analytical reflection coe ffi cients given byEqs. (14)-(15). Note that these equations were derived for purelysinusoidal and monochromatic waves but the initial packet iswell represented by a dominant frequency (wavelength), and thisis the reason of the agreement with the analytical results. Although we are mainly interested in the fast reflection prob-lem, for completeness the slow mode reflection problem is alsostudied. The di ff erence with respect to the fast MHD reflectionproblem is that reflection of the slow MHD mode at the bound-ary generates a fast MHD mode. The velocity components forthis problem are V x = ¯ S I ω − k c c k x k S e i ( ω t + k x x + k S z ) − ¯ S R ω − k c c k x k S e i ( ω t + k x x − k S z ) + ¯ F G e i ( ω t + k x x − k F z ) , (25) V z = ¯ S I e i ( ω t + k x x − k S z ) + ¯ S R e i ( ω t + k x x − k S z ) − ¯ F G c k x k F ω − k c e i ( ω t + k x x − k F z ) . (26)Applying rigid boundary conditions the following coe ffi cientsfor the reflected slow mode and the generated fast mode are Fig. 3.
Time evolution of the propagating disturbance given byEqs. (23)-(24). The continuous line represents the v x component whilethe v z component is plotted with a dotted line (its amplitude has beenmultiplied by a factor 40 for visualisation purposes). Fast and slowmodes travelling to the left are denoted as F I and S I , F I + and S I + moveto the right, and S G is the generated wave at z = z / a = ka = π/ . v / v A = k x a = . c s / v A = .
2. In this plot the time has been normalised to the character-istic time scale, τ A = v A / a . found¯ S R ¯ S I = (cid:16) ω − k c (cid:17) k F + (cid:16) ω − k c (cid:17) k S (cid:16) ω − k c (cid:17) k F − (cid:16) ω − k c (cid:17) k S , (27)¯ F G ¯ S I = c k x (cid:16) ω − k c (cid:17) (cid:16) ω − k c (cid:17) k S (cid:16) ω − k c (cid:17) k F − (cid:16) ω − k c (cid:17) k S . (28)It is worth to note that the reflection coe ffi cient of the slow modeis exactly the same as the reflection coe ffi cient of the fast modegiven by Eq. (14).
4. Standing waves
Once we understand the reflection of a propagating fast and apropagating slow wave at a rigid boundary we extend our anal- erradas et al.: Coupling between fast and slow MHD waves 5 ysis to a more realistic physical situation, i.e., the standing fastwave problem that is commonly reported by TRACE.There are several ways to study this problem, and the per-turbation method is one of them. Under this approach, we knowthat for zero- β we have a pure standing fast wave, with only a v x velocity component, that satisfies the boundary conditions.When β is di ff erent from zero but small, a v z component is in-troduced, due to Eq. (10), and even worse, this component doesnot satisfy the boundary condition. In order to compensate, wehave to add to the system a slow wave perturbation (with a dom-inant v z in comparison with v x ), which combined with the v z ofthe fast, will satisfy the boundary conditions jointly. Now the v x introduced by the slow mode is irrelevant since it is higher orderin β (which is assumed to be small).Another way to analyse this problem is to solve the fullstanding problem. This was already done by Oliver et al. (1992)and Goedbloed & Halberstadt (1994) and it is based on the su-perposition of the fast reflection problem plus the slow reflec-tion problem. The perturbation scheme and the full eigenmodeproblem give similar results, except where the solutions arecompletely mixed, i.e., when the slow mode component is notsmall in comparison with the fast component. This takes placearound the “avoided crossings” in the dispersion diagram (seeOliver et al., 1992, for further details).The advantage of the perturbation scheme is that we canstill obtain simple approximations for the velocity and densitychanges around the footpoints. Let us assume that an initialdisturbance mostly excites the standing pattern in the v x com-ponent. This standing wave is composed by the superpositionof two identical propagating waves travelling in opposite direc-tions. We concentrate on the fundamental mode, having a maxi-mum at z = L / V the amplitudeof the incoming fast wave is V / V / | v z | max ≃ c q v − c V , (29)and (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ρ ρ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) max ≃ c s q v − c V . (30)These very simple expressions give the order of the maximumvelocity and density perturbations associated to the slow re-flected mode as a function of the transverse velocity amplitude atthe loop apex and the characteristic speeds of the configuration.When c s ≪ v A the velocity perturbation reduces to | v z | max ≃ c v V = γβ V , (31)while the density perturbation is (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ρ ρ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) max ≃ c s v V = γβ V c s . (32) It is worth to calculate the order of magnitude of velocity anddensity fluctuations around the footpoint. Let us take typicalvalues: v A =
800 km s − , c s =
200 km s − , V =
80 km s − .Thus, the velocity perturbation according to Eq. (29) is | v z | max ≃ . − . Using Eq. (30) we find that the density fluctuationassociated to the slow mode is | ρ /ρ | max ≃ . ffi -cult since the slow modes interfere and the velocity and densitychanges can be larger than those given by Eqs. (29)-(30). It isalso important to remark that the density perturbation associatedto the fast standing mode has the same profile as the velocity (the v x component), therefore the fast density perturbation is zero atthe footpoints (and maximum at the apex), this explains why itis enough to consider only the density changes associated to theexcited slow modes around the footpoints. Now the time-dependent problem is solved. The initial perturba-tion, representing the excitation of a mainly fast standing wave,has the following profile, v x ( z , t = = v sin ( kz ) , (33) v z ( z , t = = . (34) Line-tying conditions are applied at the two loop footpoints. Wechoose the longitudinal wavenumber that excites the fundamen-tal standing fast MHD mode, i.e., k = π/ L . The results, repre-sented in Figure 4 manifest the generation of slow modes at thefootpoints (top panel). The fast mode drives slow modes whichmove in the direction of the loop apex, located at z = L / t ≃ L / c s ) the interfer-ence between the opposite propagating slow waves is produced.Since the two slow waves are identical but travelling in oppositedirections a quasi-standing pattern is visible (see the profile of v z around z = a in the bottom panel). Eventually, slow modesreach the opposite footpoint and get reflected. Under such con-ditions the inverted process takes place, i.e., the slow mode gen-erates a fast mode. Nevertheless, this problem would take us toofar and it is out of the scope of this work. However, it is worthto note that depending on the equilibrium and wave parametersthe generated slow modes might match the frequency of a stand-ing slow eigenmode of the loop. This means that since the slowmode is driven by the fast standing mode, the frequencies of theslow standing and fast standing waves will be basically the sameand the modes will have a highly mixed nature. This takes placearound the “avoided crossings” in the dispersion diagram. From the previous results we might think that the generation ofslow modes takes place only when rigid boundary conditions areimposed. In order to address this question, instead of line-tyingconditions, a density transition between the corona and the pho-tosphere is considered. As density profile we use the following
Terradas et al.: Coupling between fast and slow MHD waves
Fig. 4.
Time evolution of the standing fast excitation given byEqs. (33)-(34). The continuous line represents the v x component whilethe v z component is plotted with a dotted line (its amplitude has beenmultiplied by a factor 10 for visualisation purposes). For this simulation ka = π/ v / v A = k x a = . c s / v A = . idealised model, ρ = (cid:16) ρ ph − ρ c (cid:17) e − ( zw ) + (cid:16) ρ ph − ρ c (cid:17) e − ( z − Lw ) + ρ c . (35)The parameter w measures the width of the transition betweenthe photosphere, with density ρ ph , and corona, with density ρ c .The density contrast is set to ρ ph /ρ c = . Non-reflecting condi-tions are imposed at the boundaries, now placed below the pho-tosphere. The details of the model used to represent the transi-tion photosphere-corona are not important since our focus is onthe generated slow modes. The equilibrium gas pressure is set tothe same constant value in the corona and photosphere to havemagnetohydrostatic equilibrium.The results of the time-dependent simulation are plotted inFigure 5. There are some di ff erences with respect to the be-haviour found for the perfectly reflecting boundaries. For exam-ple, the system now allows the energy to escape through the pho-tosphere, i.e., there is a transmitted fast (and slow wave). Thismeans that the energy can leak through the photosphere but thisis not visible in Figure 5 because the amplitude is very small. Fig. 5.
Time evolution of the standing fast excitation given byEqs. (33)-(34) with the density model given by Eq. (35). The continuousline represents the v x component while the v z component is plotted witha dotted line (its amplitude has been multiplied by a factor 10 for visu-alisation purposes). The vertical dashed lines represent the locations ofthe transition between the corona and photosphere. For this simulation ka = π/ v / v A = k x a = . c s / v A = . ρ ph /ρ c = , w / a = . Nevertheless, the primary result here is that a similar slow modegeneration in comparison with the case of reflecting boundariesis produced in the coronal part (compare Figures 4 and 5). Theresults are qualitatively similar, thus, the overall conclusion isthat nonuniformity causes the coupling between fast and slowmodes in a similar way as line-tying conditions.
5. Link with observations
It is interesting to look for signatures of the mode coupling stud-ied in this work in the observations. Recently, Verwichte et al.(2010), have reported, using combined observations fromTRACE and EIT / SoHO, intensity fluctuations at one loop foot-point with basically the same period, around 40 min, as the fasttransverse oscillation of the loop. This long periodicity indicatesthat the origin of these density oscillations is most probably notphotospheric. Verwichte et al. (2010) suggest that the intensity erradas et al.: Coupling between fast and slow MHD waves 7 oscillations are due to variations in the line of sight column depthproduced by the changes in the loop inclination as it oscillateswith the transverse kink mode. This would explain the coinci-dence in periods, and also in the damping times.According to our study an alternative explanation for theintensity variations reported by Verwichte et al. (2010) is donein terms of the coupling between fast and slow waves due tothe line-tying conditions. The similarity of the periods of thetransverse mode and the intensity fluctuations is explained byour model where the excited slow wave, and the correspond-ing intensity oscillation, has the same periodicity as the trans-verse loop motion. Verwichte et al. (2010) show intensity varia-tions associated with a transverse oscillation in a large loop of690 ±
60 Mm length. The oscillation, which is a horizontally po-larised fundamental kink mode, has a period of P = ± v ph = ±
50 km s − . Fora thin loop in the long wavelength limit, this phase speed tendsto the kink speed, c k , of a cylindrical tube. If the intensity oscil-lation is associated with a slow magnetoacoustic mode with thesame periodicity as the kink mode, then its wavelength is equalto λ s = c s P = ( c s P / v ph )2 L . For a temperature between 0.9 and 1.8MK, the coronal sound speed is in the range 144-198 km s − .Hence, λ s = (0.30 ± L . For a standing mode, this translatesinto a wavenumber n = v ph / c s = L /λ s = ± | ρ /ρ | max = ξ , can be modelled by aten degree inclination of the loop. As the loop has a height of 236Mm, this means that ξ =
41 Mm. Hence, the velocity amplitude V = ξ ω =
106 km s − . Using Eq. (30), and the estimated value forthe Alfv´en speed, v A = ±
40 km s − , we find | ρ /ρ | max = ± L = ± v ph = ±
100 km s − , V = ± − and v A = ± − . For the peak temperature of the EUV 171Å bandpass of0.9 MK, c s =
144 km s − . The slow mode would have a wavenum-ber n = ± λ s = ±
10 Mm. Again, using Eq. (30), wecalculate | ρ /ρ | max = ff erence with respectto the case studied by Verwichte et al. (2010) is that the density perturbations are located around the loop top instead of the loopfootpoints. Fig. 6.
Relative profile of intensity variations seen in a transverselyoscillating loop studied by Verwichte et al. (2010) using TRACE as afunction of distance along the loop. The solid circle is the measurementof the intensity variation at the loop foot point. The long-dashed anddashed curves are functions -0.13sin(3 π/ L ) and -0.13sin(5 π/ L ), respec-tively, representing a possible third or fifth standing harmonic of theslow mode.
6. Discussion and conclusions
In this work we have investigated the possible e ff ects of linearcoupling between fast and slow modes that takes place due to thereflection at the photosphere. We have shown, by solving the re-flection problem, that the transverse motion of the loop producesslow MHD waves as long as the fast wave is obliquely incidenton the boundary. These slow waves are manifested as propagat-ing density fluctuations with the same frequency as the standingtransverse mode and are generated at the loop footpoints. Thewavelength of these slow modes is basically given by k s ≃ ω/ c s ,and since v A > c s their wavelength is smaller than that of thecorresponding fast standing transverse mode. Moreover, we havederived using the reflection coe ffi cient of the slow mode, simpleanalytical expressions that relate the transverse displacement atthe loop top with the amplitude of the density fluctuations pro-duced at the footpoints. The coupling between fast and slow isproportional to the plasma- β , indicating that under coronal con-ditions it will be weak in general. However, it is proportional tothe amplitude of oscillation of the fast mode, which can be largein some cases, and close to the footpoints, i.e., close to the pho-tosphere, the ratio between the sound and Alfv´en speeds can in-crease. Ideally, from the properties of the reflected slow modes,we could have indirect information about the real line-tying con-ditions in coronal loops and realistic values of the sound speednear the footpoints.The generated slow waves can eventually form a standingpattern along the loop. Thus, mode coupling might provide anexcitation mechanism of standing slow modes, di ff erent from thedriven photospheric origin usually emphasised in the literature.However, since the sound speed is assumed to be smaller thanthe Alfv´en speed, the time required for the slow mode to travelalong the loop and reflect at the footpoint is much larger than thatof the fast MHD mode. This means that slow standing modeswill need much more time than fast standing modes to build up. Terradas et al.: Coupling between fast and slow MHD waves
Nevertheless, if the wavelength of the slow modes is very shortthey might be damped by thermal conduction before the standingwave is formed.It has been shown that in the observations analysed byVerwichte et al. (2010) there are possible fingerprints of the cou-pling between fast and slow modes due to line-tying, providingan alternative explanation to the e ff ects of integration along theline of sight proposed by these authors. However, in the observa-tions studied by Verwichte et al. (2009) the mechanism is unableto explain the intensity oscillations. One of the reasons mightbe that the density changes observed in this event are reportedaround the loop apex, while the estimations are based on prop-agating waves around the footpoints. In the case of the standingpattern the density fluctuations can be much larger. Further anal-ysis of other observations will help to test the linear coupling asan operative mechanism in coronal loops.It is worth to mention that the mode coupling discussedin the work is a purely linear e ff ect. It is unrelated to thenonlinear coupling between fast and slow modes due to theponderomotive force (see Hollweg, 1971; Rankin et al., 1994;Terradas & Ofman, 2004), or the parametric coupling studied byZaqarashvili et al. (2002, 2005). It is also di ff erent from modeconversion that takes place when c s = v A . Line-tying boundaryconditions are the responsible ingredients of the coupling, butwe have shown that they are not the only way in which fast andslow waves may couple. A sharp transition between the coronaand the photosphere produces the generation of slow modes, i.e.,a change in the properties of the medium, for example the tran-sition region, leads inevitably to the coupling.Our theoretical model is based on a Cartesian geometry with-out a density enhancement representing a loop. For this reason,it is necessary to improve the model by including a slab or acylinder that mimics the e ff ect of a dense magnetic tube. Thiswill complicate the mathematical problem of the reflection atthe boundary, and it might be di ffi cult to derive simple analyti-cal expressions. For example, in a cylindrical model, the eigen-functions of the slow and fast MHD waves have di ff erent radialdependencies and might share the same azimuthal wavenumber(playing basically the role of k x in the present work). This issuewill be addressed in a future study. Acknowledgements.
J.T. acknowledges the Universitat de les Illes Balears fora postdoctoral position and the financial support received from the SpanishMICINN and FEDER funds (AYA2006-07637). J.T. thanks Ram´on Oliver andRoberto Soler for useful comments and suggestions. J.A. is supported byan International Outgoing Marie Curie Fellowship within the 7th EuropeanCommunity Framework Programme. J.A. also acknowledges support by theFund for Scientific Research - Flanders. E.V. acknowledges financial supportfrom the UK Engineering and Physical Sciences Research Council (EPSRC)Science and Innovation award.
References
Aschwanden, M. J., de Pontieu, B., Schrijver, C. J., & Title, A. M. 2002,Sol. Phys., 206, 99Aschwanden, M. J., Fletcher, L., Schrijver, C. J., & Alexander, D. 1999, ApJ,520, 880De Moortel, I. 2009, Space Science Reviews, 149, 65Erd´elyi, R. & Taroyan, Y. 2008, A&A, 489, L49Goedbloed, J. P. & Halberstadt, G. 1994, A&A, 286, 275Hollweg, J. V. 1971, J. Geophys. Res., 76, 5155Mariska, J. T. 2005, ApJ, 620, L67Mariska, J. T. 2006, ApJ, 639, 484Nakariakov, V. M., Ofman, L., Deluca, E. E., Roberts, B., & Davila, J. M. 1999,Science, 285, 862Oliver, R., Ballester, J. L., Hood, A. W., & Priest, E. R. 1992, ApJ, 400, 369Rankin, R., Frycz, P., Tikhonchuk, V. T., & Samson, J. C. 1994, J. Geophys. Res.,99, 21291 Schrijver, C. J., Aschwanden, M. J., & Title, A. M. 2002, Sol. Phys., 206, 69Schrijver, C. J. & Brown, D. S. 2000, ApJ, 537, L69Stein, R. F. 1971, ApJS, 22, 419Terradas, J. & Ofman, L. 2004, ApJ, 610, 523Vasquez, B. J. 1990, ApJ, 356, 693Verwichte, E., Aschwanden, M. J., Van Doorsselaere, T., Foullon, C., &Nakariakov, V. M. 2009, ApJ, 698, 397Verwichte, E., Foullon, C., & Van Doorsselaere, T. 2010, ApJ, 717, 458Wang, T., Innes, D. E., & Qiu, J. 2007, ApJ, 656, 598Wang, T. J., Solanki, S. K., Curdt, W., et al. 2003a, A&A, 406, 1105Wang, T. J., Solanki, S. K., Innes, D. E., Curdt, W., & Marsch, E. 2003b, A&A,402, L17Zaqarashvili, T. V., Oliver, R., & Ballester, J. L. 2002, ApJ, 569, 519Zaqarashvili, T. V., Oliver, R., & Ballester, J. L. 2005, A&A, 433, 357
Appendix A:
We use the linearised MHD equations of the momentum, induc-tion, energy and continuity equation. Assuming Fourier analysisin the x − direction we have: ∂ v x ∂ t = ρ − ik x p − ik x B µ b z + B µ ∂ b x ∂ z ! , (A.1) ∂ v z ∂ t = − ρ ∂ p ∂ z , (A.2) ∂ b x ∂ t = B ∂ v x ∂ z , (A.3) ∂ b z ∂ t = − B ik x v x , (A.4) ∂ p ∂ t = − γ p ik x v x + ∂ v z ∂ z ! , (A.5) ∂ρ ∂ t = − ρ ik x v x + ∂ v z ∂ z ! . (A.6)In these equations B is the equilibrium magnetic field and ρ and p the equilibrium density and gas pressure, respectively. Therest of the variables correspond to the perturbed magnitudes. Toeliminate the imaginary complex numbers we simply define v ∗ x = i v x , (A.7) b ∗ x = i b x . (A.8)With this transformation the time-dependent equations aresolved using standard numerical techniques.The dispersion relation, given by Eq. (1), is easily derivedassuming in the previous equations a temporal dependence ofthe form e i ω t and a spatial dependence with the z − coordinate ofthe form e ik z z . We have used the following standard definitionsfor the Alfv´en and sound speeds, v A = B √ µρ , (A.9) c s = r γ p ρ .ρ .