Fast magnetohydrodynamic oscillations in a multifibril Cartesian prominence model. A. J. Díaz, R. Oliver, and J. L. Ballester

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1 A&A 440, (2005) DOI: / : c ESO 2005 Astronomy & Astrophysics Fast magnetohydrodynamic osciations in a mutifibri Cartesian prominence mode A. J. Díaz, R. Oiver, and J. L. Baester Departament de Física, Universitat de es Ies Baears, Pama de Maorca, Spain e-mai: antonio@mcs.st-and.ac.uk;[ramon.oiver;dfsjb0]@uib.es Received 25 January 2005 / Accepted 5 May 2005 Abstract. Observations of quiescent fiaments show very fine structures which suggests that they can be composed of smascae threads or fibris. Two-dimensiona, high-resoution observations point out that individua fibris or groups of fibris may osciate independenty with their own periods. In this paper, we study the fast magnetohydrodynamic modes of osciation of mutifibri Cartesian systems to represent the osciations of the fibri structure of a rea prominence. In the case of a system made of equa fibris, our resuts show that the ony non-eaky mode is the symmetric one, which means that a the fibris osciate in spatia phase with the same frequency. On the other hand, in a system made of non-equa fibris, i.e. with different Afvén speeds, the resuts show that the ampitudes of osciation are higher in the denser fibris, that the frequency of osciation of the ony non-eaky mode is sighty smaer than that of the dominant fibri considered aone, and that a the fibris aso osciate in phase. Key words. Sun: osciations Sun: magnetic fieds Sun: corona Sun: prominences Sun: fiaments 1. Introduction High-resoution observations of quiescent fiaments show very fine structures within its body, suggesting that they are composed of sma-scae threads. The existence of this interna structure in prominences was suggested by Menze & Evans (1953), was carified with the improvement of observationa capabiities (Engvod 1976; Engvod et a. 1987) and evidence for the existence of horizonta fine structures within prominences was ater found (Schmieder & Mein 1989; Schmieder et a. 1991; Lin et a. 2005). For instance, Engvod et a. (1987) observed a quiescent fiament and, from the study of the prominence-corona interface, deduced that the fine structure of the coo fiament core may consist of thin magnetic fux ropes oriented at an ange of 20 to the fiament ong axis. Taking into account the observationa evidence, magnetostatic equiibrium modes for prominence fibris have been constructed by Baester & Priest (1989), Degenhardt & Deinzer (1993) and Schmitt & Degenhardt (1996). These modes represent a prominence fibri by means of a hot-coo oop modeed using the thin fux tube approximation. On the other hand, the existence of sma ampitude, periodic veocity osciations in quiescent soar prominences is a we-known phenomenon (see Engvod 2001; Oiver & Baester 2002, for thorough reviews), and two-dimensiona, high-resoution observations (Yi et a. 1991; Yi & Engvod 1991; Engvod 2001; Lin 2004) have reveaed that individua fibris or groups of fibris may osciate independenty with Present address: Mathematica Institute, University of St. Andrews, St. Andrews, KY16 9SS, Scotand, UK. their own periods, which ie between 3 and 20 min. Hence, one of the basic questions in prominence seismoogy is whether periodic changes in prominences are aways associated with their fibri structure or not. The first theoretica investigation of periodic prominence perturbations taking into account the prominence fine structure was performed using Cartesian geometry by Joarder et a. (1997). Díaz et a. (2001, hereafter Paper I) have done a more in-depth anaytica and numerica study of this type of configuration and the most important concusions extracted from this study are that prominence fibris can ony support a few modes of osciation (those with the smaest frequency, since high harmonics cannot be trapped inside the thin oop) and that the spatia structure of the fundamenta even and odd kink modes is such that the veocity ampitude outside the fibri is sti significant over ong distances, i.e. the energy confinement in the fibri is rather poor. An interesting consequence is that fibris may actuay osciate in groups rather than individuay. However, the study in Paper I ony considered a singe, isoated fibri, thus negecting the interaction among fibris that shoud be expected in a rea prominence. Motivated by the observations suggesting that groups of fibris may osciate together, our main goa in this paper is to study the interactions between osciating fibris. Foowing the same procedure as in Paper I, we use Cartesian geometry, athough we are aware that a more accurate description of prominence fibris requires the use of cyindrica geometry. The paper is organised as foows: in Sect. 2 the equiibrium mode, the basic assumptions, the fast wave equations as we Artice pubished by EDP Sciences and avaiabe at or

2 1168 A. J. Díaz et a.: Fast MHD osciations in mutifibri Cartesian systems ( 2 ρ 0 t 2 ) ( ) 2 c2 T v z 2 z + c2 s pt = 0, (3) c 2 z t f Fig. 1. The equiibrium configuration used in this study of a system of two fibris. The grey zone represents the cod part of the sabs, i.e. the prominence. The density in the prominence region is ρ p, in the evacuated (corona) part of the sab, ρ e, and in the corona environment, ρ c. The magnetic fied is uniform and parae to the z-axis, and the whoe configuration is invariant in the y-direction. as their anaytica soution are described. In Sect. 3 we present and discuss the resuts obtained when a coupe of fibris are considered, whie in Sect. 4 the numerica soution for the case of severa fibris is considered. In Sect. 5 our concusions are presented. 2. Mode, basic equations and anaytica soution The equiibrium mode consists of a coection of fibris separated a distance 2 (which is the reevant parameter and can differ between pairs when more that two fibris are considered). Each fibri is then modeed as in Paper I: a straight sab of tota ength 2L made of a cod and dense part (the prominence fibri itsef) with ength 2W and density ρ p, and a hotter, corona gas with density ρ e occupying the rest of the sab. The thickness of the structure is 2b and it is embedded in the corona environment, with density ρ c. The sab is anchored in the photosphere, so its footpoints are subject to ine-tying conditions. Finay, the pasma is permeated by a uniform magnetic fied directed aong the prominence fibri. Because gravity is negected, a other physica variabes (ρ, T and p) are aso uniform in each of the three regions. We aso assume invariance in the y-direction. A sketch of the mode for two fibris is potted in Fig Perturbation wave equations To derive a wave equation for this mode we foow the procedure expained in Díaz et a. (2002). We consider a uniform, static pasma with unperturbed density ρ 0 and equiibrium magnetic fied B 0 = B 0 ê z. Next, inear, adiabatic perturbations about this equiibrium are introduced and the magnetic fied and pressure perturbations, B and p, are eiminated in favour of the veocity and the tota pressure perturbations, u and p T, so the foowing equations are obtained (Roberts 1991), p T = ρ 0 c 2 v z t A z ρ 0c 2 f u, (1) ( 2 ρ 0 t 2 ) p T 2 c2 A u z 2 + = 0, t (2) where the symbo stands for the components of the perturbed veocity and the gradient perpendicuar to B 0, c s is the sound speed, c A the Afvén speed and the other characteristic speeds and c 2 T = c 2 s + c 2 A. are defined as c 2 f = c2 s + c2 A Next, we consider the imit β 0, which impies that the sow mode is absent: the component of the perturbed veocity aong the magnetic fied, v z, is identicay zero from Eq. (3). Using p T as main dependent variabe we derive the foowing equation, ( 2 t 2 c2 A 2 ) p T = 0, (4) which is soved together with Eq. (2), necessary for some of the boundary conditions. We are ony interested in stationary perturbations, so in what foows a tempora dependence of the form e iωt is considered Anaytica soution for two identica fibris The aim is to sove Eqs. (2) and (4) with the appropriate boundary conditions for the two-fibri system sketched in Fig. 1. The standard method for soving them in a cosed region is separation of variabes, p T (x, z) = u(x)h(z), which gives us two ordinary differentia equations (since ongitudina propagation is not considered), d 2 dx u(x) = 2 λ2 u(x), (5) d2 dz + ω2 h(z) = λ 2 h(z). 2 (6) c 2 A These equations must be soved in three regions (numbered 1, 2 and 3, respectivey): the corona environment ( x > 2b + ), the fibris ( > x > 2b + ) and the region between them ( x < ), whose physica properties are supposed to be identica to the corona ones. Therefore, our soution can be written as (x) h(1) (z), x 2b +, n=1 n=1 u (1) n p T (x, z) = u (2) n (x) h (2) n (z), x < 2b +, n=1 n u (3) n (x) h (3) n (z), 0 x <, in which the need to use a the basis functions to fufi the boundary conditions on the fibri surfaces has been taken into account (see Paper I for a discussion of this issue). Now, the z-dependent functions of Eq. (6) are cacuated, taking into account the ine-tying at z = ±L and the jump conditions at z = ±W (for the sab region). Notice aso that the corona medium is the same in regions 1 and 3, so h (1) n (z) = h (3) n (z), and using the same properties of Paper I the inner basis functions can be expressed in terms of the outer ones in the form h (2) n (z) = m=1 H nm h (1) n (z), in which the coefficients H nm are the ones cacuated in Paper I. (7)

3 A. J. Díaz et a.: Fast MHD osciations in mutifibri Cartesian systems 1169 Since a system made of equa fibris is under consideration, the equations can be soved in the region x 0and0 z L, so there are even and odd modes in the z-direction and even and odd modes in the x-direction. The even modes in the x- direction are those in which both fibris osciate in phase and hereafter we ca them symmetric modes, whie the odd ones are antisymmetric modes. Taking into account these symmetries, the soution of the x-dependent part for a symmetric mode is of the form u (1) n (x) = A n e λ(1) n (x 2b ), x 2b +, u (2) n (x) = B sin n sin λ (2) n (x ) n (x ), x < 2b +, +B cos n cos λ (2) u (3) n (x) = C n cosh λ (3) n x, 0 x <. The z-dependent part depends on the Afvén speed profie aong the fibri. Foowing the mode in Edwin & Roberts (1982), we coud take a cod sab where these functions are just h (1) n (z) = h (2) n (z) = h (3) n (z) = L 1/2 cos π(2n 1)z/(2L), and H nm = δ nm, so the modes of two cod Cartesian sabs coud be studied, but here we wi work directy with our mode, and these functions are the same as in Paper I for the corona (numbered 1 and 3 here) and the fibri (numbered 2). The next step is to appy the boundary conditions at x = and x = 2b +, that is, continuity of the tota pressure and of the x-components of the perturbed veocity and magnetic fied, namey n [u] = 0, n [B] = 0, [p T ] = 0. (9) It is usefu to deduce an expression of the component of the perturbed veocity in the x-direction from Eq. (2): v x (x,y,z) = iω ρ 0 c 2 A 1 λ 2 n=1 n (8) du n (x) h n (z). (10) dx Operating with these expressions the coefficients A n and C n in Eq. (8) can be eiminated, and we obtain for the symmetric modes =0 =0 H n [ B cos λ (3) n sinh λ(3) n Bsin λ (2) n cosh λ(3) H n [ λ (1) n (B sin sin λ (2) n 2b + B cos cos λ (2) n 2b) n ] = 0, + λ (2) n (B sin cos λ (2) n 2b B cos sin λ (2) n 2b) ] = 0, (11) and simiar expressions for the antisymmetric ones. The above equations constitute two infinite systems of homogeneous agebraic equations for the coefficients B sin and B cos, with coefficients that depend on ω. The condition to have a non-trivia soution is that the determinant of this system vanishes, providing us with the dispersion reation from which the eigenvaues of the probem can be obtained. Obviousy, an infinite determinant is extremey difficut to treat, so it is truncated by taking B s = 0forn > N. Thus, we are eft with a 2N-order determinant, obtained by using the first N basis functions ony (with N arge, at east 24). Once the eigenfrequencies have been obtained, a the other perturbed quantities in the probem can be determined. This system ony has a non-trivia soution if some of the λ (2) n are aowed to become compex. In this case, the x- dependence inside the fibri is in the form of hyperboic functions instead of trigonometric ones in Eq. (11). The soution method deveoped in this subsection can aso be appied with minor changes to other systems, such as two non-equa fibris or more than two of them. However, the resuting system of equations is much more compicated and difficut to treat than Eqs. (11). For that reason, the finite-differencesnumerica code described in Arregui et a. (2001, 2003) has been used. It soves the resuting partia differentia wave equations for the perturbed veocity, athough the perturbed pressure can aso be obtained as an output. The agreement with the anaytica soution is extremey good in the probem of two fibris and, moreover, the numerica program reproduces even the step discontinuities in the soution due to the MHD boundary conditions. 3. Two-fibri system There is no reason to assume that a the fibris composing a prominence are identica. In fact, observations of prominences suggest that they are highy non-homogeneous in density and temperature, whie the possibe inhomogeneity in magnetic fied is much more difficut to confirm. However, first of a, we can study the interaction among identica fibris conforming a homogeneous prominence to investigate the new features in the most simpe exampe. Then, a simiar study wi be numericay performed for the case of non-identica fibris i.e. for a non-homogeneous prominence Identica fibris: dependence of the modes on the fibri separation For this study, the key parameter is, haf the distance between the two fibris. Notice that in the imit,twoseparate fibris are recovered (and both the sausage and kink modes from Paper I are present), whie in the imit 0 the fibris coaesce and form a singe one with width 4b and, as before, the sausage and kink modes corresponding to this situation are present. First of a, in Fig. 2 the variation of the frequency for symmetric and antisymmetric modes is shown for two different sets of parameters. From Paper I it can be easiy checked that the imits and 0 are correcty described, and when the fibris are separated by more than twice their width ( > 2b approximatey) there is no couping between them, i.e. the symmetric and antisymmetric modes are equivaent in frequency and correspond to those of a singe fibri osciating aone. When the two fibris are coser, the interaction is ceary shown and the two modes become separated in frequency. In the imit 0 the frequency of the symmetric modes is smaer than that of the antisymmetric ones and, because of the interaction, the modes can cross the cut-off frequency. As an exampe, in Fig. 2a, the fourth symmetric mode, which is under the cutoff frequency for a fibri having b/l = 0.5 (imit 0) but

4 1170 A. J. Díaz et a.: Fast MHD osciations in mutifibri Cartesian systems Fig. 2. Two identica fibris: variation of the frequency with the haf-separation for symmetric even modes (soid ines) and antisymmetric even modes (dotted ines) for two different sets of parameters: a) W/L = 0.1, ρ e /ρ c = 0.6, ρ p /ρ c = 200 and b/l = 0.5; b) W/L = 0.1, ρ e /ρ c = 1, ρ p /ρ c = 1000 and b/l = 0.5 (ony the 10 first eigenfunctions in each symmetry). Notice that in a) the fourth symmetric mode appears from the cut-off (dashed ine) when the fibris get coser. Fig. 3. Two identica fibris: variation of frequency with the haf-separation for symmetric even modes (soid ines) and antisymmetric even modes (dotted ines) for two sets of parameters: a) W/L = 0.1, ρ e /ρ c = 0.6, ρ p /ρ c = 200 and b/l = 0.005; b) W/L = 0.1, ρ e /ρ c = 1, ρ p /ρ c = 1000 and b/l = Now there is ony one surviving mode in each symmetry. Notice that the antisymmetric mode has a cut-off (dashed ine) for ow vaues of b/l. is over the cut-off for a singe fibri of width b/l = 0.25 (imit ), appears for ow vaues of the parameter and becomes eaky when is increased. On the other hand, note the compex interaction of modes shown in Fig. 2b (arger density contrast, ρ p /ρ c ). A mode can cross other modes with different types of symmetry due to the couping, but not modes having the same symmetry. The spacing between the modes is not reguar, because there are space coupings for this set of parameters (see Paper I). It is aso noticeabe in both Figs. 2a and 2b that the higher the order of the mode, the ater the interaction between symmetric and antisymmetric modes starts. The expanation for this behaviour is that even in a singe fibri, the modes with higher orders are more spatiay confined, so the fibris must be coser to fee their mutua interaction. The mentioned above features ony appear when the thickness of the fibris is far from the reaistic range b/l For this situation, the behaviour is iustrated in Fig. 3. There are few confined modes eft and decreasing the parameter b/l there woud be ony one eft with its frequency tending to π/2 (see Paper I). In this case, the infuence of both fibris on the frequency is noticeabe even for high vaues of the separation 2/L compared to the fibri width b/l (in our case, over 500 times), so in this mode two prominence fibris woud aways interact. Furthermore, the antisymmetric mode becomes eaky when the two fibris become coser, as can be ceary seen in Fig. 3. Hence, the system is constrained to osciate in phase if reaistic vaues of the parameters are used Identica fibris: spatia profies We now investigate the spatia properties of the modes. The soution for the x-component of the perturbed veocity is potted in Fig. 4, where the fibris are wide and the interaction is strong enoughto differentiate very we symmetrica and antisymmetrica modes in the spatia structure (and even in frequency). It can aso be checked that the dependence in the z-coordinate is neary the same as for one fibri aone (described in Paper I),

5 A. J. Díaz et a.: Fast MHD osciations in mutifibri Cartesian systems 1171 (a) (b) Fig. 4. Two identica fibris: a) symmetric; and b) antisymmetric first mode for two wide fibris with parameters W/L = 0.1, ρ e /ρ c = 0.6, ρ p /ρ c = 200, b/l = 0.5 (Fig. 2) and haf separation /L = 0.5. In the pane z = 1.1 a sketch of the equiibrium mode has been incuded, with the fibri region shaded in grey. Notice that the whoe range in the z direction has been potted, and aso that the two fibris are incuded in these surface pots. Fig. 5. Two identica fibris: cuts of v x in the direction z = 0 of the first symmetric (soid ine) and antisymmetric (dashed ine) mode for the parameters W/L = 0.1, ρ e /ρ c = 0.6, ρ p /ρ c = 200, b/l = 0.01 (Fig. 3) and a) /L = 0.1, b) /L = 1, c) /L = 2andd) /L = 5. Notice that the x-scae is changed in the ast two panes, whie in the first one, the sausage mode has passed through a cut-off (i.e. has become eaky). so we wi concentrate on the features concerning the spatia profies across the fibris. Let us move now to reaistic parameters. Figure 5 dispays the evoution of the spatia structure of the two surviving modes when the distance between the fibris is increased (ony the region x 0 is represented, since it is easy to reconstruct the whoe soution due to the symmetry of the modes). In Fig. 5a, ony the symmetric mode is present, so both fibris osciate in spatia phase. In Fig. 5b, the antisymmetric mode is aso present, but just under the cut-off frequency, and therefore it extends to distances far away from the system. In Fig. 5c, the two modes become coser in frequency and aso in spatia shape, and finay in Fig. 5d, they are quite simiar. From Fig. 5d it is aso cear how the singe fibri modes are recovered. Adding these modes, the ampitude in the second fibri (that corresponds to x/l = 5, not shown in the graph)

6 1172 A. J. Díaz et a.: Fast MHD osciations in mutifibri Cartesian systems subsections is ost), but for reaistic vaues of the thickness and separation, the interaction is very strong, so the fibris end up osciating in phase with veocity perturbations far away from them, with minor changes from the identica fibris case. 4. Numerica anaysis of mutifibri systems 4.1. Homogeneous prominence Fig. 6. Dispersion reation for two non-identica fibris for the set of parameters W/L = 0.1, ρ e /ρ c = 0.6, b/l = and ρ p /ρ c = 200 for one of them and ρ p /ρ c = 150 for the other. The soid and dashed ines are the symmetric and antisymmetric modes respectivey. vanishes, and what is eft is just the first one osciating on its own (with a tiny contribution of the second one). If the second fibri was pushed further away, the singe fibri resuts woud be recovered exacty Non-identica fibris We now investigate the eigenmodes of non-identica fibris; this is done numericay in order to avoid cumbersome cacuations. The difference between the fibris is their density, which modifies the Afvén veocity within the fibri. The first difference from the case of identica fibris is that in the imit of high separation there are not two degenerate modes (symmetric and antisymmetric), but two fibris osciating aone and each fibri having its own eigenfrequency. As the fibris come coser the modes interact, and the mode coming from the denser fibri becomes the one in which both fibris vibrate in phase (simiar to a symmetric mode), whie the mode coming from the ess dense fibri becomes the mode in which both fibris vibrate in opposition of phase (simiar to an antisymmetric mode). For a reaistic vaue of the fibri thickness, the antisymmetric mode becomeseaky when the fibri separation is reduced, especiay for the vaues that might be expected in soar prominences. This behaviour is iustrated in Fig. 6. Regarding the spatia structure, it can be seen in Fig. 7 that for the symmetric mode (Fig. 7a) the perturbed veocity is higher in the denser fibri, whie for the antisymmetric mode (Fig. 7b) it is higher in the ess dense one. When the fibris are separated by a distance of the order of their thickness the spatia structure is simiar to that of a singe fibri (Fig. 8a) osciating with a frequency sighty smaer than that of the dense fibri. However, if we zoom on the fibri region (Fig. 8b) we see that there are sma bumps in the regions of dense materia. Therefore, if the fibris are not equa, for ong separations between them there are important differences from the system of identica fibris: the modes do not tend to a singe frequency but to the vaues corresponding to the fibris aone. Their spatia structure is aso different, with veocity perturbations mainy in one of the fibris (because the symmetry of the previous The anaytica method described in Sect. 2.2 can be extended with no theoreticadifficuty to the study of systems with more than two fibris. However, the system of equations akin to Eq. (11) becomes too cumbersome and, therefore, we numericay sove the partia differentia equation for the perturbed veocity component. For simpicity, we first study systems of n f identica fibris having thickness 2b, with a separation 2 between consecutive fibris. With these conditions, and in the imit of no interaction (/L arge enough) there are as many modes with the same frequency as fibris, and for 0 there is a singe osciating fibri with thickness n f 2b (both imits can be cacuated with the techniques of Paper I). The resuting dispersion reations for four fibris can be seen in Fig. 9. The main concusion of this graph is that for reaistic separations of fibris (that is, b) there is ony one non-eaky mode (the first symmetric one, in which a the fibris osciate in phase), whie the modes with fibris osciating in opposition become eaky for arge vaues of. Moreover, the frequency of the confined mode is very cose to that of a singe fibri with width n f 2b (for the figure, the resuts in Paper I give ω(b/l = 0.02) = c A /L, which is the numerica vaue of Fig. 9 in the imit /L 0). Regarding the spatia structure, the perturbed veocity component around the fibri region is potted in Fig. 10 for the system of four identica fibris. This pane shows the veocity spatia structure in and near the fibris and, since the interaction is very strong (/L sma), the goba shape approaches that for a singe fibri having width n f 2b. Thus, the overa perturbation is simiar to the one in Fig. 8a, with a peak in the prominence region and a sow decay away from it, but if we zoom on that region a the fibris are marked as a sma bumps, as can be seen in Fig. 10, but they osciate with just sighty more ampitude than the surrounding corona medium Non-homogeneous prominence The above mode can be appied to more compex configurations such as those expected to be found in rea fiaments. As an exampe, we study the osciatory modes of the structure potted in Fig. 11, in which a the fibris are inside fux tubes such as those described in Fig. 1. The different fibri density ratios represent the density inhomogeneity of a rea prominence, i.e. adifferent Afvén veocity for each fibri, and the thickness and separation between fibris have been chosen randomy within a reaistic range. We obtain the eigenmodes of this configuration. When the separation between fibris is that shown in Fig. 11, there is a strong interaction between them since the perturbation can

7 A. J. Díaz et a.: Fast MHD osciations in mutifibri Cartesian systems 1173 Fig. 7. Cuts in the direction z = 0 for two non-identica fibris for the set of parameters W/L = 0.1, ρ e /ρ c = 0.6, b/l = 0.005, /L = 1.25 and density ρ p /ρ c = 200 for the one situated at x/l = 1.25 and ρ p /ρ c = 150 for the one at x/l = Pane a) is a symmetric mode with ωl/c A = 1.412; and pane b) an antisymmetric mode with ωl/c A = 1.525, which is near the cut-off frequency. Fig. 8. a) Cut in the direction z = 0 for the symmetric mode (ωl/c A = 1.317) of two non-identica fibris for the set of parameters W/L = 0.1, ρ e /ρ c = 0.6, b/l = 0.005, /L = 0.01, ρ p /ρ c = 200 for one situated at x/l = 0.01 and ρ p /ρ c = 150 for the one at x/l = b) Detai of the zone around the fibris (shaded regions). Fig. 9. Logarithmic pot of the dispersion reation for four identica fibris for the set of parameters W/L = 0.1, ρ e /ρ c = 0.6, ρ p /ρ c = 200, b/l = The soid and dashed ines are symmetric and antisymmetric modes respectivey. Fig. 10. Cut in the direction z = 0 for the first symmetric mode for the parameters W/L = 0.1, ρ e /ρ c = 0.6, ρ p /ρ c = 200, b/l = and /L = = 3b/L for a system with four fibris. A sma x-range has been seected for a better rendition of the structure around the fibris (shaded regions).

8 1174 A. J. Díaz et a.: Fast MHD osciations in mutifibri Cartesian systems Fig. 11. Sketch of a cut in the direction z = 0 of the normaised density profie (ρ/ρ c ) in the case of an non-homogeneous mutifibri system. The normaised numerica vaues of the density are printed in each fibri, whie the corona medium has a vaue of 1. The fibri width and the separation between them are aso marked. Fig. 13. Logarithmic pot of the dispersion reation for the mutifibri system of Fig. 11. The range in the frequency axis has been seected smaer to better dispay the detais. Notice that now we cannot identify modes in phase and in opposition of phase directy (except the fundamenta one, which aways corresponds to a the fibris osciating in phase, see main text). Fig. 12. Detai of the zone around the fibris in a cut in the direction z = 0 for the symmetric mode (ωl/c A = 1.407) of the system of non-identica fibris presented in Fig. 11. The fibri regions have been shadowed. easiy overcome their separation and, as a resut, there is ony one even non-eaky mode: the one with a the fibris osciating in phase. The overa structure is ike the one in Fig. 8a, whie the fine structure around the fibris is that potted in Fig. 12. The spatia profie of the veocity is simiar to that of the densest fibri osciating aone, with sma contributions from the ess dense fibris. Now we study the frequency dependence on the fibri separation. We have chosen to increase the fibri separation (named ref ) keeping the widths of the fibris and the ratios of separation between them as in Fig. 12. The resuts are potted in Fig. 13 and compared with the singe dominant fibri osciating aone (dotted ine). In the imit ref, the structure and the frequencies of each fibri osciating aone are recovered, whie for sma vaues of ref a the non-phase osciating modes are eaky, and ony the mode described before remains, having a sighty smaer frequency than in the case of the singe dominant fibri mode. Aso, ooking at the spatia structure of the modes for separation ref, we find that the spatia structure of each mode tends to be that of fibris osciating aone with their own frequency such as described in Paper I. For the first mode, a the fibris osciate in phase; for the second mode there are two groups of fibris osciating with opposition of phase between them; simiary, for the third mode there are three groups, for the fourth mode ony two fibris osciate in phase (and the rest in opposition of phase), and finay, for the fifth mode a the fibris vibrate in opposition of phase with respect to their immediate neighbours. However, a these modes, except the fundamenta one described in Fig. 12, are eaky in the reaistic range ( ref b). Therefore, for reaistic vaues of the separation between fibris, the mutifibri system osciates in phase, with simiar ampitudes and the same frequency (smaer than the eigenfrequency of the densest fibri), so if we ooked at it with a broader scae, we woud see something simiar to Fig. 8a. Moreover, there woud not be any trace of other fibri eigenmodes, at east in the stationary state. 5. Concusions Using the anaytica and numerica procedure deveoped in Paper I, we have constructed a very simpe mode to study the interaction between fibris when fast MHD are excited within them. The anaytica resuts have been checked numericay in the simpest case, and both resuts are in a perfect agreement. With the goa to mimic as accuratey as possibe rea prominences, two different situations have been studied: homogeneous prominences, composed of fibris with identica physica properties, and non-homogeneous prominences, in which the density of the considered fibris is different, i.e. the Afvén veocity is the parameter being modified. When considering identica fibris, the symmetric and antisymmetric modes of osciation have been described, and the genera resut is that when two or more fibris are considered, a the modes except the first symmetric one become eaky for reaistic vaues of fibri separation. This fact has an important consequence: if a fibri is perturbed and starts to osciate, ony the stationary modes described in Paper I remain after a suitabe time scae. However, in a system of fibris, the symmetric and antisymmetric modes woud be excited but, at the end, ony the symmetric mode woud remain, i.e. the fibris woud osciate with the same frequency. Then, extrapoating these resuts

9 A. J. Díaz et a.: Fast MHD osciations in mutifibri Cartesian systems 1175 to the whoe prominence, after some time, the prominence fibris woud osciate in spatia phase with the same frequency, so the prominence woud seem to be an osciating sab with its fine structure hardy discernibe, but with the characteristic frequencies refecting the fibri presence. Therefore, a group of these fibris in a soar prominence woud osciate together with a frequency just sighty different form a sab without structure and width n f 2b. If we continued this process for arger vaues of n f, we woud tend to a periodic aternance of fibris and corona regions that coud be studied with simiar techniques to the ones used to describe a periodic array of sabs in Berton & Heyvaerts (1987). This imit is eft for further work. In the case of non-identica fibris, i.e. a non-homogeneous prominence, because of symmetry breaking there are no symmetric or antisymmetric soutions, and the most important concusion is that when reaistic separations among the fibris of the system are considered, the fibris osciate in phase with sighty different ampitudes and with a frequency which is smaer than that of the densest fibri. The effect of ongitudina propagation, simiar to what was done for a singe fibri in Díaz et a. (2003), remains to be studied. It is expected that the interaction becomes weaker, but for reaistic vaues of the parameters there are ony sight changes. If the ongitudina wavenumber k y is high enough, some antisymmetric modes coud aso be trapped. On the other hand, there is the possibiity to drain some energy from the excited fibri (apart from the eaky modes not studied here) and transfer it to its partners. The resuts presented here seem to suggest that after some time a the fibris of a prominence region woud be osciating in phase with the same frequency. This coud expain why groups of fibris seem to osciate together (Lin 2004). Finay, this study needs to be extended to more reaistic geometries. It has been shown that the geometry has a very strong infuence on the structure of the eigenmodes (Díaz et a. 2002), so one must be carefu when appying these concusions to a rea soar prominence. In this regard, this work is ony a first step for further modes in this direction. A next step woud to study the interaction of two identica cyindrica fibris; for such a system of two fibris, the modes of a singe fibri woud be shifted and a new array of modes woud appear due to the ack of symmetry in the azimutha direction. Then, the coective behaviour exhibited by packed cyindrica fux tubes coud be considered and thus the resuts of the mutifibri Cartesian systems studied in this paper coud be extended. Acknowedgements. A. J. Díaz, R. Oiver and J. L. Baester acknowedge the financia support received from the Spanish MCyT under grant AYA A. J. Díaz aso thanks the Spanish MCyT for grant BFM References Arregui, I., Oiver, R., & Baester, J. L. 2001, A&A, 369, 1122 Arregui, I., Oiver, R., & Baester, J. L. 2003, A&A, 402, 1129 Baester, J. L., & Priest, E. R. 1989, A&A, 225, 213 Berton, R., & Heyvaerts, J. 1987, So. Phys., 109, 201 Degenhardt, U., & Deinzer, W. 1993, A&A, 278, 288 Díaz, A. J., Oiver, R., Erdéyi, R., & Baester, J. L. 2001, A&A, 379, 1083 (Paper I) Díaz, A. J., Oiver, R., & Baester, J. L. 2002, ApJ, 580, 550 Díaz, A. J., Oiver, R., & Baester, J. L. 2003, A&A, 402, 781 Edwin, P. M., & Roberts, B. R. 1982, So. Phys., 76, 239 Engvod, O. 1976, So. Phys., 49, 283 Engvod, O. 2001, in INTAS Workshop on MHD Waves in Astrophysica Pasmas, ed. J. L. Baester, & B. Roberts, Universitat de es Ies Baears, Spain Engvod, O., Kjedseth-Moe, O., Bartoe, J. D. F., & Brueckner, G. 1987, in Proc. 21st ESLAB Symp., ESA SP-275, 21 Joarder, P. S., Nakariakov, V., & Roberts, B. 1997, So. Phys., 173, 81 Lin, Y. 2004, Ph.D. Thesis, Univ. of Oso Lin, Y., Engvod, O., van der Voort, L. R., Wiik, J. E., & Berger, T. E. 2005, So. Phys., submitted Menze, D. H., & Evans, J. W. 1953, Convegno Vota, 11, 119 Roberts, B. 1991, in Advances in So. System Magnetohydrodynamics, ed. E. R. Priest, & A. W. Hood (Cambridge Press), 110 Schmieder, B., & Mein, P. 1989, Hvar Obs. Buetin 13, IAU Cooq., 119, 31 Schmieder, B., Raadu, M., & Wiik, J. E. 1991, A&A, 252, 353 Schmitt, D., & Degenhardt, U. 1996, in Rev. Mod. Astron., ed. G. Kare (Springer-Verag) Yi, Z., & Engvod, O. 1991, So. Phys., 134, 275 Yi, Z., Engvod, O., & Kei, S. L. 1991, So. Phys., 132, 63